%------------------------------------------------------------------------------
% File     : ITP282_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_BuildupMemImp 00337_015688
% 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    : 0093_VEBT_BuildupMemImp_00337_015688 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12462 (4399 unt;1882 typ;   0 def)
%            Number of atoms       : 25559 (10320 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 16249 (2362   ~; 491   |;1308   &)
%                                         (2785 <=>;9303  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1940 ( 850 fml;1090 var)
%            Number of X terms     : 1188 (   0  []; 910 ite; 278 let)
%            Number of types       :  381 ( 380 usr)
%            Number of type conns  : 1768 (1111   >; 657   *;   0   +;   0  <<)
%            Number of predicates  :   86 (  83 usr;   2 prp; 0-3 aty)
%            Number of functors    : 1440 (1440 usr; 412 con; 0-8 aty)
%            Number of variables   : 25511 (24656   !; 855   ?;25511   :)
% 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 18:08:44.384
%------------------------------------------------------------------------------
% Could-be-implicit typings (380)
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    fun_set_nat_o: $tType ).

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

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

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

tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_o_fun_nat_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__Numeral____Type__Onum1_J,type,
    set_Numeral_num1: $tType ).

tff(ty_n_t__Set__Oset_It__Numeral____Type__Onum0_J,type,
    set_Numeral_num0: $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__itself_It__Numeral____Type__Onum1_J,type,
    itself_Numeral_num1: $tType ).

tff(ty_n_t__itself_It__Numeral____Type__Onum0_J,type,
    itself_Numeral_num0: $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__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__Nat__Onat_Mt__Real__Oreal_J,type,
    fun_nat_real: $tType ).

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

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

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

tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    fun_rat_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__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__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__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

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

tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    fun_complex_o: $tType ).

tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
    fun_assn_o: $tType ).

tff(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
    set_literal: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
    itself_finite_1: $tType ).

tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

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

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

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

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

tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $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__Nat__Onat_J,type,
    list_nat: $tType ).

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

tff(ty_n_t__fun_It__Real__Oreal_M_Eo_J,type,
    fun_real_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Assertions__Oassn,type,
    assn: $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__Uint32__Ouint32,type,
    uint32: $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 (1502)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_complex > fun_complex_o ).

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

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

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_real_real ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_int ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_real_real ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_int_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: assn > fun_assn_assn ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

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

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_nat > fun_na7234455742697417626_nat_o ).

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

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: complex > fun_na1591161823130732966omplex ).

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

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

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_set_nat_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_set_nat_nat > fun_fu1237770272497958833_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_complex_nat > fun_fu6122667721776870753plex_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_complex_nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

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: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_assn > fun_nat_assn ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_nat_Code_integer > fun_nat_Code_integer ).

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_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_nat_assn > fun_na2454419004871237530n_assn ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: int > fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_set_nat_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_complex_nat > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: real > fun_fu4701863041303578399l_assn ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: vEBT_VEBT > fun_fu2706034715333652099T_assn ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: complex > fun_fu7489246418442965795x_assn ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: real > fun_fu4701863041303578399l_assn ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: vEBT_VEBT > fun_fu2706034715333652099T_assn ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: complex > fun_fu7489246418442965795x_assn ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_int ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_int_int ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_int > fun_fu8464300013002972271at_int ).

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

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_VEBT_VEBT > fun_fu3446001173448610095VEBT_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_VEBT_VEBT > fun_fu155073193946645565VEBT_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_Code_integer > fun_fu7672990395897251493eger_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_Code_integer > fun_fu4547929877023156982eger_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_VEBT_VEBT > fun_fu3470462164674526041VEBT_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_VEBT_VEBT > fun_fu7917602684971597757VEBT_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_complex > fun_fu2497916604042137945plex_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_complex > fun_fu6925816717872577647plex_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_Code_integer > fun_fu4523568072766896123eger_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: set_Code_integer > fun_fu7141499545569947298eger_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: set_real > fun_fu583245617705562381real_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: set_real > fun_fu6415922825302012245real_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_VEBT_VEBT > fun_fu7910578496649732265VEBT_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_VEBT_VEBT > fun_fu7091630716096207293VEBT_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_nat > fun_fu1116345184836134341_nat_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: set_nat > fun_fu3109912920687205353_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_int > fun_fu2934462529731303693_int_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_int > fun_fu607692679349591509_int_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_real_assn > fun_fu2023413033720386729l_assn ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_VEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_VEBT_VEBT_assn > fun_fu7718536751978369761T_assn ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_assn > fun_fu2792963925203051945t_assn ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_int_assn > fun_fu2048648494261043369t_assn ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_complex_assn > fun_fu2210291882803266345x_assn ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_Co4490034186337852034r_assn > fun_fu1898041513128675305r_assn ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_VEBT_VEBT_real > fun_fu4911191927098811621T_real ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_real > fun_fu4002475321494201261t_real ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_int_assn > fun_int_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_complex_assn > fun_complex_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_Co4490034186337852034r_assn > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_Co3960896333048504454r_real > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: vEBT_VEBT > fun_fu4973138925650586885T_assn ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: complex > fun_fu8961391208694906739x_assn ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: code_integer > fun_fu4755565544374692864r_assn ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: real > fun_fu5856939556780759917l_assn ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: int > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: code_integer > fun_fu8135725022974991116r_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_nat > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: vEBT_VEBT > fun_o_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_nat_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: array_VEBT_VEBTi > fun_li8825114373765265308i_assn ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_real_o ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_real_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_int_o ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_real_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_real ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_real > fun_fu6119738925743952495t_real ).

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

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

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_real_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_real_o > fun_real_fun_real_o ).

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

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_in1696471567231710611nt_int ).

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

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

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

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: set_nat > fun_nat_fun_nat_o ).

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

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

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

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

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

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oentails,type,
    entails: fun_assn_fun_assn_o ).

tff(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: fun_li3352006551604695530i_assn > assn ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: ( array_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > fun_assn_fun_assn_o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: fun_nat_o > int ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun_nat_o > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

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

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: ( nat * code_integer ) > code_integer ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

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

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

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

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001_Eo,type,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist__update_001t__Int__Oint,type,
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tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: ( list_VEBT_VEBTi * nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

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

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

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

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

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

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

tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: ( list_VEBT_VEBTi * nat ) > vEBT_VEBTi ).

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

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

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

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

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

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

tff(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: ( nat * vEBT_VEBTi ) > list_VEBT_VEBTi ).

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

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

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

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Misc_Oslice_001_Eo,type,
    slice_o: ( nat * nat * list_o ) > list_o ).

tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
    slice_int: ( nat * nat * list_int ) > list_int ).

tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
    slice_nat: ( nat * nat * list_nat ) > list_nat ).

tff(sy_c_Misc_Oslice_001t__Real__Oreal,type,
    slice_real: ( nat * nat * list_real ) > list_real ).

tff(sy_c_Misc_Oslice_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Misc_Oslice_001t__VEBT____Definitions__OVEBT,type,
    slice_VEBT_VEBT: ( nat * nat * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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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,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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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__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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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: num > nat ).

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

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

tff(sy_c_Num_Opow,type,
    pow: fun_num_fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

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__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

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_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P7891021755743879167et_nat: option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P2758821087663475997et_nat: option5899113203645146382et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P994283844982041979et_nat: produc8799533983703829792et_nat > option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
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tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

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

tff(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

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

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: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: ( option_int * option_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: fun_op493924057549976310_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: ( option_rat * option_rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: ( option_real * option_real ) > $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 > fun_se7289222521248229150eger_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > fun_set_real_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__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
    ord_less_char: ( char * char ) > $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__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_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le3534172602983796987VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_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__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: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: fun_op493924057549976310_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: ( set_VEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: ( char * char ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: ( extended_enat * extended_enat ) > extended_enat ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

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

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

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__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    top_to3689904429138878997l_num1: set_Numeral_num1 ).

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_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
    top_top_set_literal: set_literal ).

tff(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > fun_nat_assn ).

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

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: ( heap_e7401611519738050253t_unit * set_nat ) > produc3658429121746597890et_nat ).

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__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc6084888613844515218_VEBTi: ( vEBT_VEBT * vEBT_VEBTi ) > produc3625547720036274456_VEBTi ).

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc6017135503889078682et_nat: ( fun_Pr6460944699146269935_nat_o * produc3658429121746597890et_nat ) > produc8799533983703829792et_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc7291855603632349880et_nat: ( fun_Pr6460944699146269935_nat_o * produc8799533983703829792et_nat ) > produc3112576180687729086et_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc3653971535690468304at_nat: ( fun_Pr2028404973800837934_nat_o * produc6121120109295599847at_nat ) > produc3739950874144486240at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7814092408141220007at_nat: ( fun_Pr2477239776880934807at_nat * produc6121120109295599847at_nat ) > produc1260151622200656695at_nat ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( fun_na4138050610385705875at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

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

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

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

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

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_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_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > fun_complex_complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: ( rat * rat ) > rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Assertions__Oassn,type,
    dvd_dvd_assn: fun_assn_fun_assn_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > fun_complex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: $o > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_o_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: fun_o_real ).

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

tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: fun_nat_int > int ).

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: fun_nat_nat > nat ).

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

tff(sy_c_Series_Osummable_001t__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_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Code_integer_o > set_Code_integer ).

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
    collect_list_o: fun_list_o_o > set_list_o ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    collec3483841146883906114nteger: fun_li1325434182413301700eger_o > set_li6976499617229504675nteger ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: fun_list_complex_o > set_list_complex ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: fun_list_int_o > set_list_int ).

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

tff(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
    collect_list_real: fun_list_real_o > set_list_real ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: fun_list_VEBT_VEBT_o > set_list_VEBT_VEBT ).

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

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

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_o > set_set_nat ).

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

tff(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    image_4470545334726330049nteger: ( fun_Co8499238673658658103nteger * set_Code_integer ) > set_Code_integer ).

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

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

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

tff(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    image_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * set_VEBT_VEBT ) > set_nat ).

tff(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: ( $o * set_o ) > set_o ).

tff(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
    insert_Code_integer: ( code_integer * set_Code_integer ) > set_Code_integer ).

tff(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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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__Num__Onum,type,
    insert_num: ( num * set_num ) > set_num ).

tff(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: ( rat * set_rat ) > set_rat ).

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

tff(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > set_VEBT_VEBT ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Assertions__Oassn,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( fun_na7078387940644931716nteger * nat * nat * code_integer ) > code_integer ).

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

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

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat * 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__Code____Numeral__Ointeger,type,
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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__Code____Numeral__Ointeger,type,
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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_OatLeastLessThan_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
    set_or4029947393144176647an_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
    set_or66887138388493659n_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or3540276404033026485et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
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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__Code____Numeral__Ointeger,type,
    set_or4266950643985792945nteger: ( code_integer * code_integer ) > set_Code_integer ).

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_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: int > fun_int_int ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > fun_int_int ).

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

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

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    syntax7398250324933576852n_assn: fun_assn_fun_assn_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: ( assn * heap_Time_Heap_nat * fun_nat_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

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

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

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

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

tff(sy_c_Uint32_OUint32,type,
    uint322: fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( $o * $o ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr3720955412185915546EBTi_o ).

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 ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_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: fun_VE6480751511521678896_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: fun_VE6480751511521678896_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: fun_VE6480751511521678896_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__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_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__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001_Eo,type,
    vEBT_L7363604446928714179sn_o_o: ( fun_o_fun_o_assn * list_o * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( fun_o_fun_nat_assn * list_o * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( fun_o_fun_real_assn * list_o * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    vEBT_L4260503343685368993omplex: ( fun_co6163879640491479806x_assn * list_complex * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Int__Oint,type,
    vEBT_L134985006839036959ex_int: ( fun_co6164383388265235324t_assn * list_complex * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    vEBT_L137475477348087235ex_nat: ( fun_co1748589094918083360t_assn * list_complex * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    vEBT_L2479436891206192927x_real: ( fun_co8000059489268409724l_assn * list_complex * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8524933119956041985T_VEBT: ( fun_co2031555705788908698T_assn * list_complex * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L6066640139021943271_int_o: ( fun_int_fun_o_assn * list_int * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001_Eo,type,
    vEBT_L7887682484454631235_nat_o: ( fun_nat_fun_o_assn * list_nat * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( fun_na448437756743965086l_assn * list_nat * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( fun_real_fun_o_assn * list_real * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    vEBT_L2162147798726695391omplex: ( fun_VE496882281059548826x_assn * list_VEBT_VEBT * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( fun_VE123837653852366616t_assn * list_VEBT_VEBT * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    aa_fun1332161875654541722_nat_o: ( fun_fu1237770272497958833_nat_o * fun_set_nat_nat ) > fun_set_nat_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun8193769468198689164VEBT_o: ( fun_fu6329627000648617163VEBT_o * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J,type,
    aa_fun8737571391773076682T_assn: ( fun_fu7718536751978369761T_assn * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    aa_fun3597767889293385422T_real: ( fun_fu4911191927098811621T_real * fun_VEBT_VEBT_o ) > fun_VEBT_VEBT_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun5142337294461892954VEBT_o: ( fun_fu7910578496649732265VEBT_o * fun_VEBT_VEBT_assn ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J,type,
    aa_fun7127563710338807932T_assn: ( fun_fu2706034715333652099T_assn * fun_VEBT_VEBT_assn ) > fun_VEBT_VEBT_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun876682331325704374VEBT_o: ( fun_fu7091630716096207293VEBT_o * fun_VEBT_VEBT_assn ) > fun_fu7910578496649732265VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J_J,type,
    aa_fun7778866309842214326T_assn: ( fun_fu4973138925650586885T_assn * fun_VEBT_VEBT_assn ) > fun_fu2706034715333652099T_assn ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun677759971260770784VEBT_o: ( fun_fu3446001173448610095VEBT_o * fun_VE5569070535936029429omplex ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun5749144123862529590VEBT_o: ( fun_fu155073193946645565VEBT_o * fun_VE5569070535936029429omplex ) > fun_fu3446001173448610095VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun5889122065779655690VEBT_o: ( fun_fu3470462164674526041VEBT_o * fun_VEBT_VEBT_rat ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun1978406156034293430VEBT_o: ( fun_fu7917602684971597757VEBT_o * fun_VEBT_VEBT_rat ) > fun_fu3470462164674526041VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_fun1965331558590768990VEBT_o: ( fun_fu6610162456291744429VEBT_o * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    aa_fun1825138017709814396T_real: ( fun_fu8198533418994508931T_real * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun5637434397837810486VEBT_o: ( fun_fu8270842325377534013VEBT_o * fun_VEBT_VEBT_real ) > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    aa_fun910392650362481850T_real: ( fun_fu4141963935195952137T_real * fun_VEBT_VEBT_real ) > fun_fu8198533418994508931T_real ).

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

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 > fun_se7289222521248229150eger_o ).

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

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

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

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

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

tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $o ).

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

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

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

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

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

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_x____,type,
    x: list_VEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: array_VEBT_VEBTi ).

tff(sy_v_xb____,type,
    xb: vEBT_VEBTi ).

tff(sy_v_xc____,type,
    xc: vEBT_VEBTi ).

% Relevant facts (10578)
tff(fact_0_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_1_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ).

% assnle
tff(fact_2_False,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,na))) ).

% False
tff(fact_3_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))) ).

% div2_Suc_Suc
tff(fact_4_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(P),pure_assn((B)))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ) ).

% ent_pure_pre_iff
tff(fact_5_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,numeral_numeral_nat(bit0(one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_6_power__odd__eq,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_7_power__odd__eq,axiom,
    ! [A: assn,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) = aa_assn_assn(times_times_assn(A),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),Nb)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_8_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,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_9_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,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_10_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,numeral_numeral_nat(bit0(one))),Nb))) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_11_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(times_times_assn(pure_assn((A))),pure_assn((B))) = pure_assn(( (A)
        & (B) )) ).

% merge_pure_star
tff(fact_12_power__even__eq,axiom,
    ! [A: assn,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),Nb)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_13_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,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_14_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,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_15_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_16_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_17_power__even__eq,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),numeral_numeral_nat(bit0(one))) ).

% power_even_eq
tff(fact_18_Suc__double__not__eq__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_19_double__not__eq__Suc__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_20_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_21_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_22_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).

% times_divide_eq_left
tff(fact_23_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_24_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_25_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_26_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_27_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_28_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).

% divide_divide_eq_right
tff(fact_29_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_30_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_31_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).

% times_divide_eq_right
tff(fact_32_power4__eq__xxxx,axiom,
    ! [Xd: assn] : aa_nat_assn(power_power_assn(Xd),numeral_numeral_nat(bit0(bit0(one)))) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_33_power4__eq__xxxx,axiom,
    ! [Xd: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(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,Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_34_power4__eq__xxxx,axiom,
    ! [Xd: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),numeral_numeral_nat(bit0(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,Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_35_power4__eq__xxxx,axiom,
    ! [Xd: int] : aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(bit0(one)))) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(times_times_int(Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_36_power4__eq__xxxx,axiom,
    ! [Xd: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(bit0(one)))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_37_power4__eq__xxxx,axiom,
    ! [Xd: complex] : aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(bit0(one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Xd),Xd)),Xd)),Xd) ).

% power4_eq_xxxx
tff(fact_38_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( pure_assn((P)) = pure_assn((Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_39_power__mult__numeral,axiom,
    ! [A: nat,M: 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),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_40_power__mult__numeral,axiom,
    ! [A: real,M: 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),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_41_power__mult__numeral,axiom,
    ! [A: int,M: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_42_power__mult__numeral,axiom,
    ! [A: complex,M: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_complex(power_power_complex(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_43_power__mult__numeral,axiom,
    ! [A: code_integer,M: num,Nb: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_44_power__mult__numeral,axiom,
    ! [A: assn,M: num,Nb: num] : aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M))),numeral_numeral_nat(Nb)) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% power_mult_numeral
tff(fact_45_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_46_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_47_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(times_times_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_48_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_49_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ) ).

% even_Suc
tff(fact_50_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))) ) ) ).

% even_Suc_div_two
tff(fact_51_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))) ) ) ).

% odd_Suc_div_two
tff(fact_52_dvd__power__same,axiom,
    ! [Xd: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xd),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_53_dvd__power__same,axiom,
    ! [Xd: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_54_dvd__power__same,axiom,
    ! [Xd: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xd),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).

% dvd_power_same
tff(fact_55_dvd__power__same,axiom,
    ! [Xd: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xd),Y)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xd),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).

% dvd_power_same
tff(fact_56_dvd__power__same,axiom,
    ! [Xd: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_57_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_58_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_59_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(divide_divide_int(A),B)),Nb) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% div_power
tff(fact_60_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),numeral_numeral_nat(K))),numeral_numeral_nat(L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_61_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_62_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(bit0(Nb))) ).

% even_numeral
tff(fact_63_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_num_int(numeral_numeral_int,bit0(Nb))) ).

% even_numeral
tff(fact_64_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B2) ) ).

% evenE
tff(fact_65_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B2) ) ).

% evenE
tff(fact_66_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ~ ! [B2: int] : A != aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B2) ) ).

% evenE
tff(fact_67_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_68_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_69_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,bit0(one))),A)
     => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_70_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),R) = aa_assn_assn(times_times_assn(P),aa_assn_assn(times_times_assn(Q),R)) ).

% assn_times_assoc
tff(fact_71_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_assn_assn(times_times_assn(Q),P) ).

% assn_times_comm
tff(fact_72_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Q),R)
       => aa_assn_o(aa_assn_fun_assn_o(entails,P),R) ) ) ).

% ent_trans
tff(fact_73_ent__refl,axiom,
    ! [P: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,P),P) ).

% ent_refl
tff(fact_74_ent__iffI,axiom,
    ! [A2: assn,B3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B3)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B3),A2)
       => ( A2 = B3 ) ) ) ).

% ent_iffI
tff(fact_75_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_76_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_77_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_78_divide__divide__times__eq,axiom,
    ! [Xd: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xd),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xd),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_79_divide__divide__times__eq,axiom,
    ! [Xd: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_80_divide__divide__times__eq,axiom,
    ! [Xd: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(Xd,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xd),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_81_times__divide__times__eq,axiom,
    ! [Xd: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xd),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_82_times__divide__times__eq,axiom,
    ! [Xd: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_83_times__divide__times__eq,axiom,
    ! [Xd: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(Xd,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xd),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_84_power__commuting__commutes,axiom,
    ! [Xd: assn,Y: assn,Nb: nat] :
      ( ( aa_assn_assn(times_times_assn(Xd),Y) = aa_assn_assn(times_times_assn(Y),Xd) )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(Xd),Nb)),Y) = aa_assn_assn(times_times_assn(Y),aa_nat_assn(power_power_assn(Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_85_power__commuting__commutes,axiom,
    ! [Xd: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xd) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_86_power__commuting__commutes,axiom,
    ! [Xd: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xd),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xd) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),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,Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_87_power__commuting__commutes,axiom,
    ! [Xd: int,Y: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xd),Y) = aa_int_int(times_times_int(Y),Xd) )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xd),Nb)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(power_power_int(Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_88_power__commuting__commutes,axiom,
    ! [Xd: code_integer,Y: code_integer,Nb: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Xd) )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_89_power__commuting__commutes,axiom,
    ! [Xd: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xd),Y) = aa_complex_complex(times_times_complex(Y),Xd) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xd),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xd),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_90_power__mult__distrib,axiom,
    ! [A: assn,B: assn,Nb: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(times_times_assn(A),B)),Nb) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),Nb)),aa_nat_assn(power_power_assn(B),Nb)) ).

% power_mult_distrib
tff(fact_91_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_92_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_93_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(times_times_int(A),B)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).

% power_mult_distrib
tff(fact_94_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),Nb) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ).

% power_mult_distrib
tff(fact_95_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_mult_distrib
tff(fact_96_power__commutes,axiom,
    ! [A: assn,Nb: nat] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),Nb)),A) = aa_assn_assn(times_times_assn(A),aa_nat_assn(power_power_assn(A),Nb)) ).

% power_commutes
tff(fact_97_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_98_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_99_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_100_power__commutes,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_commutes
tff(fact_101_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_102_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_divide
tff(fact_103_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_104_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),Nb) = divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_105_power__mult,axiom,
    ! [A: nat,M: 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,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),Nb) ).

% power_mult
tff(fact_106_power__mult,axiom,
    ! [A: real,M: 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,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),Nb) ).

% power_mult
tff(fact_107_power__mult,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),Nb) ).

% power_mult
tff(fact_108_power__mult,axiom,
    ! [A: complex,M: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),Nb) ).

% power_mult
tff(fact_109_power__mult,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),Nb) ).

% power_mult
tff(fact_110_power__mult,axiom,
    ! [A: assn,M: nat,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),M)),Nb) ).

% power_mult
tff(fact_111_div__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),Nb)),Q2) ).

% div_mult2_eq
tff(fact_112_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),P2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Q),Q3)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(P),Q)),aa_assn_assn(times_times_assn(P2),Q3)) ) ) ).

% ent_star_mono
tff(fact_113_power__Suc2,axiom,
    ! [A: assn,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,Nb)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),Nb)),A) ).

% power_Suc2
tff(fact_114_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_115_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_116_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_117_power__Suc2,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),A) ).

% power_Suc2
tff(fact_118_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_119_power__Suc,axiom,
    ! [A: assn,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,Nb)) = aa_assn_assn(times_times_assn(A),aa_nat_assn(power_power_assn(A),Nb)) ).

% power_Suc
tff(fact_120_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_121_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_122_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_123_power__Suc,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_Suc
tff(fact_124_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_125_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_126_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),collect_VEBT_VEBT(P))
    <=> aa_VEBT_VEBT_o(P,A) ) ).

% mem_Collect_eq
tff(fact_127_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_128_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_129_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_130_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_131_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_132_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_133_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_134_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_135_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_136_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_137_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P,X)
        <=> aa_nat_o(Q,X) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_138_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(Q,X) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_139_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X: complex] :
          ( aa_complex_o(P,X)
        <=> aa_complex_o(Q,X) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_140_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_nat_o(P,X)
        <=> aa_set_nat_o(Q,X) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_141_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(bit0(Nb))),numeral_numeral_nat(bit0(one))) = numeral_numeral_nat(Nb) ).

% numeral_Bit0_div_2
tff(fact_142_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,bit0(Nb))),aa_num_int(numeral_numeral_int,bit0(one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_143_power2__eq__square,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(bit0(one))) = aa_assn_assn(times_times_assn(A),A) ).

% power2_eq_square
tff(fact_144_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_145_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_146_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) = aa_int_int(times_times_int(A),A) ).

% power2_eq_square
tff(fact_147_power2__eq__square,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) ).

% power2_eq_square
tff(fact_148_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),numeral_numeral_nat(bit0(one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_149_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_150_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_151_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_152_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_153_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_154_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_155_div2__even__ext__nat,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),numeral_numeral_nat(bit0(one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Xd)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Y) )
       => ( Xd = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_156_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_157_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_158_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),numeral_numeral_nat(bit0(one))) ) ) ) ).

% bit_eq_rec
tff(fact_159_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B) )
        & ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(divide_divide_int(B),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ) ).

% bit_eq_rec
tff(fact_160_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(one)),Nb) = bit0(Nb) ).

% num_double
tff(fact_161_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != bit0(Nb) ).

% semiring_norm(83)
tff(fact_162_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one ).

% semiring_norm(85)
tff(fact_163_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(W)),Z)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_164_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_165_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_166_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_167_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_168_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_169_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_170_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_171_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_172_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_173_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb)) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_174_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% numeral_times_numeral
tff(fact_175_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),B) ).

% pow_sum
tff(fact_176_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xd: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_vebt_buildup(Nb)),Xd) ).

% buildup_nothing_in_leaf
tff(fact_177_even__odd__cases,axiom,
    ! [Xd: nat] :
      ( ! [N: nat] : Xd != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xd != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_178_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_179_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_180_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_181_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numeral_numeral_nat(M) = numeral_numeral_nat(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_182_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_183_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( bit0(M) = bit0(Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_184_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_185_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_186_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : plus_plus_rat(numeral_numeral_rat(V),plus_plus_rat(numeral_numeral_rat(W),Z)) = plus_plus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_187_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_188_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_189_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_190_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_191_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_192_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb)) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_193_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% numeral_plus_numeral
tff(fact_194_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_195_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(B,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_196_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_197_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_198_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_199_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_200_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_201_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_202_semiring__norm_I13_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(M)),bit0(Nb)) = bit0(bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% semiring_norm(13)
tff(fact_203_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_204_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_205_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,numeral_numeral_nat(bit0(one))),D))),L) ).

% bit_concat_def
tff(fact_206_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(B,C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),C)) ).

% distrib_left_numeral
tff(fact_207_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_208_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_209_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_210_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),C)) ).

% distrib_left_numeral
tff(fact_211_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_212_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B)),numera6620942414471956472nteger(V)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),numera6620942414471956472nteger(V))) ).

% distrib_right_numeral
tff(fact_213_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_214_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_215_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),numeral_numeral_rat(V)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_216_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),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),numeral_numeral_nat(V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),numeral_numeral_nat(V))) ).

% distrib_right_numeral
tff(fact_217_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_218_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(B,aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_219_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_220_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_221_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_222_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_223_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(C),A)))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_224_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(aa_rat_rat(times_times_rat(C),A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_225_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_226_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_227_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_228_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_229_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(C),A)),B))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_230_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_231_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_232_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) ) ) ).

% odd_add
tff(fact_233_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B) ) ) ).

% odd_add
tff(fact_234_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) ) ) ).

% even_add
tff(fact_235_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B) ) ) ).

% even_add
tff(fact_236_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_237_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_238_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),numeral_numeral_nat(bit0(one))) = M ).

% add_self_div_2
tff(fact_239_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_240_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% is_num_normalize(1)
tff(fact_241_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% is_num_normalize(1)
tff(fact_242_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_243_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_244_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_245_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B)),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_246_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_247_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_248_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_249_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_250_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),plus_p5714425477246183910nteger(B,C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_251_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_252_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_253_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_254_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_255_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% comm_semiring_class.distrib
tff(fact_256_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B)),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% comm_semiring_class.distrib
tff(fact_257_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% comm_semiring_class.distrib
tff(fact_258_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_259_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_260_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_261_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% distrib_left
tff(fact_262_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),plus_p5714425477246183910nteger(B,C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)) ).

% distrib_left
tff(fact_263_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% distrib_left
tff(fact_264_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_265_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_266_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_267_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% distrib_right
tff(fact_268_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B)),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% distrib_right
tff(fact_269_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% distrib_right
tff(fact_270_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : plus_plus_rat(aa_rat_rat(times_times_rat(A),E),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),C)) = plus_plus_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),E),C) ).

% combine_common_factor
tff(fact_271_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_272_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_273_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(plus_plus_int(A),B)),E)),C) ).

% combine_common_factor
tff(fact_274_combine__common__factor,axiom,
    ! [A: code_integer,E: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(A,B)),E),C) ).

% combine_common_factor
tff(fact_275_combine__common__factor,axiom,
    ! [A: complex,E: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E)),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),E)),C) ).

% combine_common_factor
tff(fact_276_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_277_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_278_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_279_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_280_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_281_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_282_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_283_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_284_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_285_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),plus_plus_rat(B,C)) ) ) ).

% dvd_add
tff(fact_286_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_287_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(plus_plus_int(B),C)) ) ) ).

% dvd_add
tff(fact_288_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_289_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_290_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_291_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(plus_plus_rat(A,B),C) = plus_plus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_292_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(bit0(Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_293_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_real(bit0(Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb)) ).

% numeral_Bit0
tff(fact_294_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_rat(bit0(Nb)) = plus_plus_rat(numeral_numeral_rat(Nb),numeral_numeral_rat(Nb)) ).

% numeral_Bit0
tff(fact_295_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_nat(bit0(Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(Nb)) ).

% numeral_Bit0
tff(fact_296_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,bit0(Nb)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_297_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_298_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_299_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_300_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_301_power__add,axiom,
    ! [A: assn,M: nat,Nb: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),M)),aa_nat_assn(power_power_assn(A),Nb)) ).

% power_add
tff(fact_302_power__add,axiom,
    ! [A: real,M: 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,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_303_power__add,axiom,
    ! [A: nat,M: 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,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_304_power__add,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_305_power__add,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_add
tff(fact_306_power__add,axiom,
    ! [A: complex,M: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_307_mult__2,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Z) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2
tff(fact_308_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_309_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_310_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Z) = plus_plus_rat(Z,Z) ).

% mult_2
tff(fact_311_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_312_mult__2,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Z) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2
tff(fact_313_mult__2__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),numera6620942414471956472nteger(bit0(one))) = plus_p5714425477246183910nteger(Z,Z) ).

% mult_2_right
tff(fact_314_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(bit0(one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_315_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_316_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(bit0(one))) = plus_plus_rat(Z,Z) ).

% mult_2_right
tff(fact_317_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_318_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2_right
tff(fact_319_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),A),B) ).

% left_add_twice
tff(fact_320_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_321_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_322_left__add__twice,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(A,B)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),A),B) ).

% left_add_twice
tff(fact_323_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_324_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),B) ).

% left_add_twice
tff(fact_325_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_326_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_327_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_328_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_329_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_330_div__exp__eq,axiom,
    ! [A: nat,M: 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,numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_331_div__exp__eq,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_332_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_333_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B)) ) ) ).

% odd_even_add
tff(fact_334_power2__sum,axiom,
    ! [Xd: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(Xd,Y)),numeral_numeral_nat(bit0(one))) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_335_power2__sum,axiom,
    ! [Xd: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_complex(power_power_complex(Y),numeral_numeral_nat(bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_336_power2__sum,axiom,
    ! [Xd: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),numeral_numeral_nat(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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_337_power2__sum,axiom,
    ! [Xd: rat,Y: rat] : aa_nat_rat(power_power_rat(plus_plus_rat(Xd,Y)),numeral_numeral_nat(bit0(one))) = plus_plus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_338_power2__sum,axiom,
    ! [Xd: 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,Xd),Y)),numeral_numeral_nat(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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_339_power2__sum,axiom,
    ! [Xd: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Xd)),Y)) ).

% power2_sum
tff(fact_340_dvdE,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A)
     => ~ ! [K2: assn] : A != aa_assn_assn(times_times_assn(B),K2) ) ).

% dvdE
tff(fact_341_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_342_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_343_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(times_times_int(B),K2) ) ).

% dvdE
tff(fact_344_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),K2) ) ).

% dvdE
tff(fact_345_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ~ ! [K2: complex] : A != aa_complex_complex(times_times_complex(B),K2) ) ).

% dvdE
tff(fact_346_dvdI,axiom,
    ! [A: assn,B: assn,K: assn] :
      ( ( A = aa_assn_assn(times_times_assn(B),K) )
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A) ) ).

% dvdI
tff(fact_347_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_348_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_349_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(times_times_int(B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_350_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_351_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A = aa_complex_complex(times_times_complex(B),K) )
     => aa_complex_o(dvd_dvd_complex(B),A) ) ).

% dvdI
tff(fact_352_dvd__def,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A)
    <=> ? [K3: assn] : A = aa_assn_assn(times_times_assn(B),K3) ) ).

% dvd_def
tff(fact_353_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_354_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_355_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(times_times_int(B),K3) ) ).

% dvd_def
tff(fact_356_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),K3) ) ).

% dvd_def
tff(fact_357_dvd__def,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
    <=> ? [K3: complex] : A = aa_complex_complex(times_times_complex(B),K3) ) ).

% dvd_def
tff(fact_358_dvd__mult,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(times_times_assn(B),C)) ) ).

% dvd_mult
tff(fact_359_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_360_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_361_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)) ) ).

% dvd_mult
tff(fact_362_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ).

% dvd_mult
tff(fact_363_dvd__mult,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),C)
     => aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(times_times_complex(B),C)) ) ).

% dvd_mult
tff(fact_364_dvd__mult2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),B)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(times_times_assn(B),C)) ) ).

% dvd_mult2
tff(fact_365_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_366_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_367_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C)) ) ).

% dvd_mult2
tff(fact_368_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ).

% dvd_mult2
tff(fact_369_dvd__mult2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
     => aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(times_times_complex(B),C)) ) ).

% dvd_mult2
tff(fact_370_dvd__mult__left,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),C) ) ).

% dvd_mult_left
tff(fact_371_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_372_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_373_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_374_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_375_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),B)),C)
     => aa_complex_o(dvd_dvd_complex(A),C) ) ).

% dvd_mult_left
tff(fact_376_dvd__triv__left,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(times_times_assn(A),B)) ).

% dvd_triv_left
tff(fact_377_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_378_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_379_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(A),B)) ).

% dvd_triv_left
tff(fact_380_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ).

% dvd_triv_left
tff(fact_381_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(times_times_complex(A),B)) ).

% dvd_triv_left
tff(fact_382_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,C: assn,D: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,C),D)
       => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A),C)),aa_assn_assn(times_times_assn(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_383_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_384_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_385_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_386_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_387_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
     => ( aa_complex_o(dvd_dvd_complex(C),D)
       => aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_388_dvd__mult__right,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),C) ) ).

% dvd_mult_right
tff(fact_389_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_390_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_391_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_392_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_393_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),B)),C)
     => aa_complex_o(dvd_dvd_complex(B),C) ) ).

% dvd_mult_right
tff(fact_394_dvd__triv__right,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A),aa_assn_assn(times_times_assn(B),A)) ).

% dvd_triv_right
tff(fact_395_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_396_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_397_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),A)) ).

% dvd_triv_right
tff(fact_398_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)) ).

% dvd_triv_right
tff(fact_399_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : aa_complex_o(dvd_dvd_complex(A),aa_complex_complex(times_times_complex(B),A)) ).

% dvd_triv_right
tff(fact_400_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(C),A)
     => ( aa_complex_o(dvd_dvd_complex(C),B)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_401_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_402_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_403_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_404_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_405_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( aa_complex_o(dvd_dvd_complex(C),A)
       => ( aa_complex_o(dvd_dvd_complex(C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_406_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_407_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_408_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_409_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_410_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_411_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B),D)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_div_div_same
tff(fact_412_mult__numeral__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(one)),A) = A ).

% mult_numeral_1
tff(fact_413_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_414_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_415_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_416_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(one)),A) = A ).

% mult_numeral_1
tff(fact_417_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_418_mult__numeral__1__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),numera6620942414471956472nteger(one)) = A ).

% mult_numeral_1_right
tff(fact_419_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_420_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_421_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_422_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(one)) = A ).

% mult_numeral_1_right
tff(fact_423_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_424_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_425_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_426_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_427_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A),C) ) ) ).

% dvd_div_mult
tff(fact_428_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_429_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_430_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ) ) ).

% div_mult_swap
tff(fact_431_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_432_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) ) ) ).

% div_mult_swap
tff(fact_433_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_434_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_435_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_436_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_437_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_438_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B),C)),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_439_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_440_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_441_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_442_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_443_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_444_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_445_mult__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc_right
tff(fact_446_add__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc_right
tff(fact_447_field__sum__of__halves,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),numeral_numeral_real(bit0(one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),numeral_numeral_real(bit0(one)))) = Xd ).

% field_sum_of_halves
tff(fact_448_field__sum__of__halves,axiom,
    ! [Xd: rat] : plus_plus_rat(divide_divide_rat(Xd,numeral_numeral_rat(bit0(one))),divide_divide_rat(Xd,numeral_numeral_rat(bit0(one)))) = Xd ).

% field_sum_of_halves
tff(fact_449_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_450_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))),numeral_numeral_nat(bit0(one))) ).

% triangle_def
tff(fact_451_high__def,axiom,
    ! [Xd: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% high_def
tff(fact_452_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xd) = 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) ) )
         => ? [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y2)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_453_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_454_mult__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc
tff(fact_455_highi__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_highi(Xd,Nb) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% highi_def
tff(fact_456_set__decode__Suc,axiom,
    ! [Nb: nat,Xd: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xd))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),numeral_numeral_nat(bit0(one))))) ) ).

% set_decode_Suc
tff(fact_457_four__x__squared,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd)),numeral_numeral_nat(bit0(one))) ).

% four_x_squared
tff(fact_458_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_459_nat_Oinject,axiom,
    ! [X2: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y22) )
    <=> ( X2 = Y22 ) ) ).

% nat.inject
tff(fact_460_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_461_semiring__norm_I6_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),bit0(Nb)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(6)
tff(fact_462_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = bit0(one) ).

% semiring_norm(2)
tff(fact_463_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_464_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,numeral_numeral_nat(Nb)) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_465_power__add__numeral2,axiom,
    ! [A: assn,M: num,Nb: num,B: assn] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M))),aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),numeral_numeral_nat(Nb))),B)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_466_power__add__numeral2,axiom,
    ! [A: real,M: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_467_power__add__numeral2,axiom,
    ! [A: nat,M: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_468_power__add__numeral2,axiom,
    ! [A: int,M: num,Nb: num,B: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),numeral_numeral_nat(M))),aa_int_int(times_times_int(aa_nat_int(power_power_int(A),numeral_numeral_nat(Nb))),B)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_469_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,Nb: num,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(Nb))),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_470_power__add__numeral2,axiom,
    ! [A: complex,M: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),numeral_numeral_nat(M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),numeral_numeral_nat(Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)))),B) ).

% power_add_numeral2
tff(fact_471_power__add__numeral,axiom,
    ! [A: assn,M: num,Nb: num] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M))),aa_nat_assn(power_power_assn(A),numeral_numeral_nat(Nb))) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_472_power__add__numeral,axiom,
    ! [A: real,M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_473_power__add__numeral,axiom,
    ! [A: nat,M: 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),numeral_numeral_nat(M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_474_power__add__numeral,axiom,
    ! [A: int,M: num,Nb: num] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A),numeral_numeral_nat(M))),aa_nat_int(power_power_int(A),numeral_numeral_nat(Nb))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_475_power__add__numeral,axiom,
    ! [A: code_integer,M: num,Nb: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(M))),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(Nb))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_476_power__add__numeral,axiom,
    ! [A: complex,M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),numeral_numeral_nat(M))),aa_nat_complex(power_power_complex(A),numeral_numeral_nat(Nb))) = aa_nat_complex(power_power_complex(A),numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% power_add_numeral
tff(fact_477_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_478_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_479_Suc__inject,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xd) = aa_nat_nat(suc,Y) )
     => ( Xd = Y ) ) ).

% Suc_inject
tff(fact_480_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_481_gcd__nat_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% gcd_nat.refl
tff(fact_482_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% gcd_nat.trans
tff(fact_483_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.eq_iff
tff(fact_484_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A)
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_485_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_486_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_487_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
          & ( B != C ) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_488_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_489_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_490_gcd__nat_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_491_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_492_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% gcd_nat.strict_implies_order
tff(fact_493_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_494_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_495_dvd__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% dvd_antisym
tff(fact_496_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,numeral_numeral_nat(V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_497_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B4: nat,C2: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B4),C2) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B4),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),C) ) ) ).

% division_decomp
tff(fact_498_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))
     => ? [B4: int,C2: int] :
          ( ( A = aa_int_int(times_times_int(B4),C2) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B4),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),C) ) ) ).

% division_decomp
tff(fact_499_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X: nat,Y2: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y2) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_500_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P3),aa_int_int(times_times_int(A),B))
     => ~ ! [X: int,Y2: int] :
            ( ( P3 = aa_int_int(times_times_int(X),Y2) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_501_add__Suc__shift,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_502_add__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc
tff(fact_503_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_504_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( M = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_505_add__mult__distrib2,axiom,
    ! [K: nat,M: 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,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_506_add__mult__distrib,axiom,
    ! [M: 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,M),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_507_high__inv,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb))),Xd)),Nb) = Y ) ) ).

% high_inv
tff(fact_508_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ) ).

% high_bound_aux
tff(fact_509_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xd) = Y )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_510_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(K)),numeral_numeral_nat(L)) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_511_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),numeral_numeral_nat(L)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_512_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(K)),numeral_numeral_nat(L)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_513_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(numeral_numeral_rat(K)),numeral_numeral_nat(L)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_514_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(K)),numeral_numeral_nat(L)) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_515_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),numeral_numeral_nat(L)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(pow,K),L)) ).

% power_numeral
tff(fact_516_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_517_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_518_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,bit0(one))),A)
     => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_519_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_520_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_521_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_522_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_523_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_524_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_525_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_526_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_527_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y22: num] :
      ( ( bit0(X2) = bit0(Y22) )
    <=> ( X2 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_528_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,bit0(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_529_lowi__def,axiom,
    ! [Xd: nat,Nb: nat] : vEBT_VEBT_lowi(Xd,Nb) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% lowi_def
tff(fact_530_list__assn__mono,axiom,
    ! [P: fun_VE905575822219544865i_assn,P2: fun_VE905575822219544865i_assn,L: list_VEBT_VEBT,L2: list_VEBT_VEBTi] :
      ( ! [X: vEBT_VEBT,X3: vEBT_VEBTi] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,X),X3)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P2,X),X3))
     => aa_assn_o(aa_assn_fun_assn_o(entails,vEBT_L6296928887356842470_VEBTi(P,L,L2)),vEBT_L6296928887356842470_VEBTi(P2,L,L2)) ) ).

% list_assn_mono
tff(fact_531_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xd: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_vebt_buildup(Nb)),Xd) ).

% buildup_nothing_in_min_max
tff(fact_532_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_533_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_534_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_535_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_536_power__one__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),one_one_nat) = A ).

% power_one_right
tff(fact_537_power__one__right,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),one_one_nat) = A ).

% power_one_right
tff(fact_538_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_539_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_540_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_541_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_542_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_543_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_544_pure__true,axiom,
    pure_assn($true) = one_one_assn ).

% pure_true
tff(fact_545_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = one_one_assn )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_546_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_547_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_548_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_549_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_550_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% numeral_less_iff
tff(fact_551_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_552_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_553_add__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add_0
tff(fact_554_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_555_add__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add_0
tff(fact_556_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y) )
    <=> ( ( Xd = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_557_add__eq__0__iff__both__eq__0,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y) = zero_zero_nat )
    <=> ( ( Xd = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_558_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_559_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_560_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(A,B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_561_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_562_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_563_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_564_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_565_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = plus_plus_rat(B,A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_566_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_567_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_568_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_569_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_570_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_571_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_572_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_573_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_574_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_575_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( plus_plus_rat(B,A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_576_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_577_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_578_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_579_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = plus_plus_rat(A,A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_580_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_581_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_582_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_583_add_Oright__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.right_neutral
tff(fact_584_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_585_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_586_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_587_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_588_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_589_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(times_times_int(zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_590_mult__zero__left,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_591_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_592_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_593_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_594_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_595_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_596_mult__zero__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_597_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_598_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_599_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_600_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_601_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_602_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_603_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_604_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_605_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_606_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_607_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_608_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_609_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_610_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_611_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_612_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_613_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_614_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_615_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_616_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_617_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_618_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_619_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_620_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_621_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_622_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_623_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_624_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_625_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_626_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_627_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_628_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_629_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_630_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_631_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_632_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_633_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_634_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_635_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_636_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_637_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_638_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_639_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_640_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_641_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_642_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_643_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_644_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_645_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_646_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_647_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_648_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_649_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_650_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_651_mult_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% mult.right_neutral
tff(fact_652_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_653_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_654_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.right_neutral
tff(fact_655_mult_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),one_one_Code_integer) = A ).

% mult.right_neutral
tff(fact_656_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_657_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_658_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% mult_1
tff(fact_659_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_660_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_661_mult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% mult_1
tff(fact_662_mult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(one_one_Code_integer),A) = A ).

% mult_1
tff(fact_663_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_664_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_665_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_666_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_667_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_668_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_669_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_670_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_671_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_672_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_673_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_674_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_675_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_676_power__one,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),Nb) = one_one_Code_integer ).

% power_one
tff(fact_677_power__one,axiom,
    ! [Nb: nat] : aa_nat_assn(power_power_assn(one_one_assn),Nb) = one_one_assn ).

% power_one
tff(fact_678_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_679_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_680_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_681_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_682_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_683_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(A),zero_zero_complex) ).

% dvd_0_right
tff(fact_684_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_685_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_686_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_687_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_688_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_689_Suc__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_690_Suc__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_eq
tff(fact_691_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_692_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_693_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_694_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_695_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_696_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_697_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_698_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_699_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_700_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_701_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_702_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_703_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_704_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_705_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_706_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_707_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_708_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_709_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_710_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_711_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_712_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_713_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_714_add__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_715_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_716_mult__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_717_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_718_mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_719_mult__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_720_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = M ) ) ).

% mod_less
tff(fact_721_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,pure_assn((B))),Q)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,one_one_assn),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_722_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_723_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_724_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),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_725_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_726_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_727_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_728_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_729_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_730_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),plus_plus_rat(B,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_731_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_732_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_733_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_734_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),plus_plus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_735_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_736_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_737_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_738_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_739_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_740_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_741_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_742_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(B,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_743_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_744_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_745_sum__squares__eq__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xd = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_746_sum__squares__eq__zero__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xd = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_747_sum__squares__eq__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y)) = zero_zero_int )
    <=> ( ( Xd = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_748_sum__squares__eq__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y)) = zero_z3403309356797280102nteger )
    <=> ( ( Xd = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_749_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_750_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_751_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_752_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_753_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_754_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_755_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_756_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(times_times_int(C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_757_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_758_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_759_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_760_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_761_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(times_times_int(B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_762_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_763_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_764_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_765_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_766_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(times_times_int(A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_767_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_768_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_769_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_770_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_real(Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_771_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_rat(Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_772_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_nat(Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_773_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_774_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_775_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = numeral_numeral_real(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_776_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = numeral_numeral_rat(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_777_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = numeral_numeral_nat(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_778_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_779_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_780_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_781_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_782_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_783_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_784_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_785_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_786_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_787_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_788_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_789_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_790_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_791_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_792_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_793_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_794_divide__self__if,axiom,
    ! [A: rat] :
      divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_795_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_796_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_797_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_798_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_799_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_800_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_801_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_802_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_803_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_804_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_805_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_806_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_807_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_808_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_809_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_810_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_811_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_812_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_813_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_814_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_815_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_816_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_817_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_818_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_819_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_820_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_821_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_822_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_823_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_824_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_825_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_826_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_827_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_828_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_829_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_830_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_831_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult1
tff(fact_832_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_833_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_834_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult2
tff(fact_835_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_836_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_837_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,divide6298287555418463151nteger(A,B)) ).

% div_mult_mult1_if
tff(fact_838_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_839_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(divide_divide_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(divide_divide_int(A),B)) ).

% div_mult_mult1_if
tff(fact_840_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_841_power__inject__exp,axiom,
    ! [A: real,M: 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),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_842_power__inject__exp,axiom,
    ! [A: rat,M: 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),M) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_843_power__inject__exp,axiom,
    ! [A: nat,M: 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),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_844_power__inject__exp,axiom,
    ! [A: int,M: 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),M) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_845_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_846_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_847_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_848_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_849_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_850_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,Nb)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_851_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),numeral_numeral_nat(K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_852_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),numeral_numeral_nat(K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_853_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),numeral_numeral_nat(K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_854_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),numeral_numeral_nat(K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_855_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),numeral_numeral_nat(K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_856_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),numeral_numeral_nat(K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_857_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_858_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_859_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_860_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_861_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_862_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_863_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_864_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_865_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_866_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_867_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_868_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_869_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_870_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_871_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_872_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_873_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_874_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_875_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_876_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_877_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_878_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_879_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_880_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_881_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_882_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_883_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_884_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_885_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_886_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_887_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_888_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_889_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_890_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_891_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_892_power__Suc0__right,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_893_power__Suc0__right,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_894_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_895_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_896_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_897_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_898_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_899_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_900_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_901_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_902_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_903_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_904_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_905_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_906_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_907_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_908_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_909_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_910_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_911_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_912_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_913_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_914_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_915_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_916_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_917_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_918_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_919_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_920_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_921_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_922_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_923_add__gr__0,axiom,
    ! [M: 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,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_924_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_925_mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_926_one__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_927_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_928_nat__power__eq__Suc__0__iff,axiom,
    ! [Xd: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xd = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_929_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = M )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_930_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_931_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_932_nat__0__less__mult__iff,axiom,
    ! [M: 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,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_933_mult__less__cancel2,axiom,
    ! [M: 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,M),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,M),Nb) ) ) ).

% mult_less_cancel2
tff(fact_934_nat__zero__less__power__iff,axiom,
    ! [Xd: 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,Xd),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_935_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_936_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_937_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_938_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_939_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_940_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_941_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_942_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),numeral_numeral_nat(Nb))
    <=> aa_num_o(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_943_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(ord_less_num(one),Nb) ) ).

% one_less_numeral_iff
tff(fact_944_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_945_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_946_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_947_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_948_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_949_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_950_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_951_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_952_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_953_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_954_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_955_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_956_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_957_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_958_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_959_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_960_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_961_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_962_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_963_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_964_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_965_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_966_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self1
tff(fact_967_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_968_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),B))),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_969_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self2
tff(fact_970_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_971_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(B),C))),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_972_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self3
tff(fact_973_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_974_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C),B)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_975_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C),A),B) = plus_p5714425477246183910nteger(C,divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self4
tff(fact_976_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_977_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),C)),A)),B) = aa_int_int(plus_plus_int(C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_978_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_979_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_980_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_981_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_982_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_983_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_984_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% Suc_1
tff(fact_985_power__strict__increasing__iff,axiom,
    ! [B: code_integer,Xd: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_986_power__strict__increasing__iff,axiom,
    ! [B: real,Xd: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_987_power__strict__increasing__iff,axiom,
    ! [B: rat,Xd: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xd)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_988_power__strict__increasing__iff,axiom,
    ! [B: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_989_power__strict__increasing__iff,axiom,
    ! [B: int,Xd: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xd)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_990_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_991_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_992_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_993_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_994_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_995_power__eq__0__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_996_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_997_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_998_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% unit_mult_div_div
tff(fact_999_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1000_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1001_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_1002_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_1003_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_1004_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: 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,M),Nb)),Nb) = M ) ) ).

% div_mult_self_is_m
tff(fact_1005_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: 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),M)),Nb) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1006_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: 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,numeral_numeral_nat(K)),Nb))),numeral_numeral_nat(K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1007_Suc__mod__mult__self1,axiom,
    ! [M: 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,M),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,M)),Nb) ).

% Suc_mod_mult_self1
tff(fact_1008_Suc__mod__mult__self2,axiom,
    ! [M: 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,M),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,M)),Nb) ).

% Suc_mod_mult_self2
tff(fact_1009_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self3
tff(fact_1010_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self4
tff(fact_1011_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% one_add_one
tff(fact_1012_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(bit0(one)) ).

% one_add_one
tff(fact_1013_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% one_add_one
tff(fact_1014_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% one_add_one
tff(fact_1015_one__add__one,axiom,
    aa_int_int(plus_plus_int(one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% one_add_one
tff(fact_1016_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1017_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1018_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1019_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1020_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1021_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1022_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(bit0(one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1023_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1024_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1025_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),numeral_numeral_nat(bit0(one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1026_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_1027_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_1028_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_1029_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_1030_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_1031_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_1032_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_1033_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A) ) ).

% even_mod_2_iff
tff(fact_1034_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A) ) ).

% even_mod_2_iff
tff(fact_1035_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A) ) ).

% even_mod_2_iff
tff(fact_1036_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(one))) ).

% mod2_Suc_Suc
tff(fact_1037_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1038_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1039_numeral__plus__one,axiom,
    ! [Nb: num] : plus_plus_rat(numeral_numeral_rat(Nb),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1040_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Nb)),one_one_nat) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1041_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_1042_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1043_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1044_one__plus__numeral,axiom,
    ! [Nb: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1045_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(Nb)) = numeral_numeral_nat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1046_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_1047_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_1048_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1049_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_1050_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1051_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_1052_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),numeral_numeral_nat(bit0(one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1053_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),numeral_numeral_nat(bit0(one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1054_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),numeral_numeral_nat(bit0(one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1055_sum__power2__eq__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one)))) = zero_zero_rat )
    <=> ( ( Xd = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1056_sum__power2__eq__zero__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = zero_zero_real )
    <=> ( ( Xd = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1057_sum__power2__eq__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( ( aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one)))) = zero_zero_int )
    <=> ( ( Xd = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1058_sum__power2__eq__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one)))) = zero_z3403309356797280102nteger )
    <=> ( ( Xd = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1059_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1060_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1061_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1062_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1063_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1064_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1065_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_1066_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_1067_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1068_mod2__gr__0,axiom,
    ! [M: 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,M),numeral_numeral_nat(bit0(one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1069_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1070_set__decode__0,axiom,
    ! [Xd: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xd))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Xd) ) ).

% set_decode_0
tff(fact_1071_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_1072_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_1073_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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)),numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_1074_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_1075_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_1076_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_1077_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_1078_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_1079_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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,bit0(one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_1080_power__less__zero__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_1081_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,numeral_numeral_nat(bit0(one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_1082_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,numeral_numeral_nat(bit0(one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_1083_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,numeral_numeral_nat(bit0(one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_1084_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(W))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1085_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),numeral_numeral_nat(W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1086_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),numeral_numeral_nat(W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1087_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),numeral_numeral_nat(W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1088_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(W)))
    <=> ( ( numeral_numeral_nat(W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1089_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),numeral_numeral_nat(W)))
    <=> ( ( numeral_numeral_nat(W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1090_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),numeral_numeral_nat(W)))
    <=> ( ( numeral_numeral_nat(W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1091_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),numeral_numeral_nat(W)))
    <=> ( ( numeral_numeral_nat(W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_1092_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1093_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1094_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,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1095_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1096_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,bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1097_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1098_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q4: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q4) ) ).

% mod_eq_0D
tff(fact_1099_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_1100_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),plus_plus_rat(A,C)) ) ) ).

% pos_add_strict
tff(fact_1101_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_1102_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% pos_add_strict
tff(fact_1103_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_1104_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,one_one_rat)) ).

% zero_less_two
tff(fact_1105_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_1106_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_1107_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1108_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_1109_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B)) ) ) ).

% add_pos_pos
tff(fact_1110_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_1111_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_pos_pos
tff(fact_1112_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1113_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1114_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1115_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1116_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_1117_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_1118_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_1119_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_1120_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_1121_nat__neq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% nat_neq_iff
tff(fact_1122_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_1123_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1124_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_1125_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_1126_gr__implies__not0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1127_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_1128_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_1129_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
              & ~ aa_nat_o(P,M3) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_1130_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                  & ~ aa_nat_o(P,M3) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_1131_linorder__neqE__nat,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( Xd != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xd) ) ) ).

% linorder_neqE_nat
tff(fact_1132_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1133_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_1134_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_1135_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_1136_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)) ) ) ).

% power_strict_increasing
tff(fact_1137_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)) ) ) ).

% power_strict_increasing
tff(fact_1138_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)) ) ) ).

% power_strict_increasing
tff(fact_1139_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)) ) ) ).

% power_strict_increasing
tff(fact_1140_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1141_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1142_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1143_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1144_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( 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),N2)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_1145_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1146_power__less__imp__less__exp,axiom,
    ! [A: real,M: 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),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1147_power__less__imp__less__exp,axiom,
    ! [A: rat,M: 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),M)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1148_power__less__imp__less__exp,axiom,
    ! [A: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1149_power__less__imp__less__exp,axiom,
    ! [A: int,M: 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),M)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1150_one__less__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% one_less_power
tff(fact_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_1161_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_1162_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_1163_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_1164_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_1165_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_1166_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),Nb) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_1167_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_1168_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_1169_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1170_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_1171_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X4) ).

% linordered_field_no_lb
tff(fact_1172_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X4) ).

% linordered_field_no_lb
tff(fact_1173_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_1174_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_1175_div__less__mono,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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,A2),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,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1176_mod__less__divisor,axiom,
    ! [Nb: nat,M: 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,M),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_1177_div__less__dividend,axiom,
    ! [Nb: nat,M: 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),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),M) ) ) ).

% div_less_dividend
tff(fact_1178_zero__reorient,axiom,
    ! [Xd: complex] :
      ( ( zero_zero_complex = Xd )
    <=> ( Xd = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_1179_zero__reorient,axiom,
    ! [Xd: real] :
      ( ( zero_zero_real = Xd )
    <=> ( Xd = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1180_zero__reorient,axiom,
    ! [Xd: rat] :
      ( ( zero_zero_rat = Xd )
    <=> ( Xd = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1181_zero__reorient,axiom,
    ! [Xd: nat] :
      ( ( zero_zero_nat = Xd )
    <=> ( Xd = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1182_zero__reorient,axiom,
    ! [Xd: int] :
      ( ( zero_zero_int = Xd )
    <=> ( Xd = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1183_one__reorient,axiom,
    ! [Xd: assn] :
      ( ( one_one_assn = Xd )
    <=> ( Xd = one_one_assn ) ) ).

% one_reorient
tff(fact_1184_one__reorient,axiom,
    ! [Xd: real] :
      ( ( one_one_real = Xd )
    <=> ( Xd = one_one_real ) ) ).

% one_reorient
tff(fact_1185_one__reorient,axiom,
    ! [Xd: rat] :
      ( ( one_one_rat = Xd )
    <=> ( Xd = one_one_rat ) ) ).

% one_reorient
tff(fact_1186_one__reorient,axiom,
    ! [Xd: nat] :
      ( ( one_one_nat = Xd )
    <=> ( Xd = one_one_nat ) ) ).

% one_reorient
tff(fact_1187_one__reorient,axiom,
    ! [Xd: int] :
      ( ( one_one_int = Xd )
    <=> ( Xd = one_one_int ) ) ).

% one_reorient
tff(fact_1188_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_1189_gr__implies__not__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1190_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1191_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1192_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1193_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_1194_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_1195_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_1196_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_1197_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(ord_less_num(A),A) ).

% verit_comp_simplify1(1)
tff(fact_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1213_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1214_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1215_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1216_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1217_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_1218_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_1219_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_1220_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_1221_linorder__neqE__linordered__idom,axiom,
    ! [Xd: real,Y: real] :
      ( ( Xd != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1222_linorder__neqE__linordered__idom,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( Xd != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xd) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1223_linorder__neqE__linordered__idom,axiom,
    ! [Xd: int,Y: int] :
      ( ( Xd != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xd) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1224_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_1225_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_1226_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_1227_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_1228_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: 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,M),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1229_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_1230_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_1231_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_1232_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_1233_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P3: nat,M: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_1234_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1235_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1236_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1237_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_1238_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_1239_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_1240_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_1241_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1242_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1243_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(ord_less_num(aa_nat_num(F,Nb)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1244_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1245_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1246_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_1247_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_1248_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_num_o(ord_less_num(aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1249_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_1250_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_1251_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_1252_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_1253_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_1254_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1255_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1256_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1257_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1258_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1259_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1260_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_1261_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,one_one_rat)),plus_plus_rat(B,one_one_rat)) ) ).

% add_mono1
tff(fact_1262_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_1263_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),one_one_int)),aa_int_int(plus_plus_int(B),one_one_int)) ) ).

% add_mono1
tff(fact_1264_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_1265_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),plus_plus_rat(A,one_one_rat)) ).

% less_add_one
tff(fact_1266_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_1267_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(A),one_one_int)) ).

% less_add_one
tff(fact_1268_less__1__mult,axiom,
    ! [M: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(M),Nb)) ) ) ).

% less_1_mult
tff(fact_1269_less__1__mult,axiom,
    ! [M: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( 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,M),Nb)) ) ) ).

% less_1_mult
tff(fact_1270_less__1__mult,axiom,
    ! [M: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),Nb)) ) ) ).

% less_1_mult
tff(fact_1271_less__1__mult,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( 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,M),Nb)) ) ) ).

% less_1_mult
tff(fact_1272_less__1__mult,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(M),Nb)) ) ) ).

% less_1_mult
tff(fact_1273_mod__Suc,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))) ).

% mod_Suc
tff(fact_1274_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1275_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1276_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1277_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_1278_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_less_numeral
tff(fact_1279_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_less_numeral
tff(fact_1280_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(Nb)) ).

% zero_less_numeral
tff(fact_1281_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_1282_add__less__zeroD,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1283_add__less__zeroD,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(Xd,Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1284_add__less__zeroD,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Xd),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1285_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1286_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1287_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1288_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1289_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_1290_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_1291_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1292_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1293_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1294_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1295_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1296_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1297_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_1298_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1299_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1300_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1301_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1302_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_1303_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1304_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1305_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1306_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1307_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1308_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1309_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1310_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1311_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1312_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_1313_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1314_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1315_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1316_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1317_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_1318_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_1319_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1320_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1321_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_1322_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_1323_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1324_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1325_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_1326_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1327_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1328_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1329_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1330_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1331_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1332_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1333_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1334_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1335_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1336_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1337_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1338_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1339_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1340_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1341_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1342_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1343_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1344_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1345_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1346_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1347_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1348_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1349_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1350_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1351_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1352_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1353_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1354_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1355_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1356_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1357_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1358_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1359_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1360_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1361_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1362_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1363_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1364_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1365_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1366_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1367_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1368_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1369_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1370_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1371_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1372_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_1373_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono
tff(fact_1374_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_1375_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_1376_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1377_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_1378_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1379_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_1380_divide__pos__pos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)) ) ) ).

% divide_pos_pos
tff(fact_1381_divide__pos__pos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_pos_pos
tff(fact_1382_divide__pos__neg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_1383_divide__pos__neg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_1384_divide__neg__pos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_1385_divide__neg__pos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_1386_divide__neg__neg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd),Y)) ) ) ).

% divide_neg_neg
tff(fact_1387_divide__neg__neg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_neg_neg
tff(fact_1388_zero__less__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% zero_less_power
tff(fact_1389_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_1390_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_1391_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_1392_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_1393_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M = aa_nat_nat(suc,J2) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1394_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_1395_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_1396_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).

% gr0_conv_Suc
tff(fact_1397_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_1398_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1399_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1400_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_1401_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_1402_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_1403_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_1404_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_1405_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_1406_power__0,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_1407_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_1408_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_1409_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1410_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_1411_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_1412_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1413_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: 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),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( M = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1414_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1415_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_1416_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_1417_nat__power__less__imp__less,axiom,
    ! [I: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_1418_nat__dvd__not__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ) ).

% nat_dvd_not_less
tff(fact_1419_dvd__pos__nat,axiom,
    ! [Nb: nat,M: 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,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_1420_mult__eq__self__implies__10,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
     => ( ( Nb = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1421_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_1422_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1423_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1424_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_1425_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_1426_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1427_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1428_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_1429_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_1430_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)) ) ).

% add_strict_right_mono
tff(fact_1431_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_1432_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_strict_right_mono
tff(fact_1433_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_1434_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)) ) ).

% add_strict_left_mono
tff(fact_1435_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_1436_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_strict_left_mono
tff(fact_1437_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_1438_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D)) ) ) ).

% add_strict_mono
tff(fact_1439_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_1440_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_strict_mono
tff(fact_1441_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_1442_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1443_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_1444_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1445_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_1446_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1447_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_1448_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1449_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_1450_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1451_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_1452_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1453_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1454_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1455_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_1456_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_1457_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(times_times_int(one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1458_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(one_one_Code_integer),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1459_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_1460_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1461_mult_Ocomm__neutral,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% mult.comm_neutral
tff(fact_1462_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_1463_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_1464_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(times_times_int(A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_1465_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),one_one_Code_integer) = A ).

% mult.comm_neutral
tff(fact_1466_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1467_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1468_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_1469_verit__sum__simplify,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1470_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_1471_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_1472_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1473_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_1474_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% add.group_left_neutral
tff(fact_1475_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_1476_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1477_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_1478_add_Ocomm__neutral,axiom,
    ! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1479_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_1480_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_1481_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1482_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_1483_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1484_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_1485_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1486_power__Suc__less,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1487_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_1488_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1489_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_1490_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_1491_power__Suc__less__one,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_1492_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_1493_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_1494_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_1495_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_1496_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J2) )
               => aa_nat_o(P,J2) ) ) ) ) ) ).

% split_mod
tff(fact_1497_dvd__power,axiom,
    ! [Nb: nat,Xd: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xd),aa_nat_rat(power_power_rat(Xd),Nb)) ) ).

% dvd_power
tff(fact_1498_dvd__power,axiom,
    ! [Nb: nat,Xd: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),Nb)) ) ).

% dvd_power
tff(fact_1499_dvd__power,axiom,
    ! [Nb: nat,Xd: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) ) ).

% dvd_power
tff(fact_1500_dvd__power,axiom,
    ! [Nb: nat,Xd: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),aa_nat_int(power_power_int(Xd),Nb)) ) ).

% dvd_power
tff(fact_1501_dvd__power,axiom,
    ! [Nb: nat,Xd: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xd),aa_nat_complex(power_power_complex(Xd),Nb)) ) ).

% dvd_power
tff(fact_1502_dvd__power,axiom,
    ! [Nb: nat,Xd: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xd = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)) ) ).

% dvd_power
tff(fact_1503_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(Nb))),numeral_numeral_nat(bit0(Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1504_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1505_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(Nb)),numera6620942414471956472nteger(bit0(Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1506_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_1507_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_1508_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_1509_dvd__mult__cancel2,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_1510_dvd__mult__cancel1,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_1511_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1512_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B)
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1513_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1514_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_1515_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_1516_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_1517_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat))) ) ).

% less_half_sum
tff(fact_1518_power__gt1__lemma,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1519_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_1520_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1521_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_1522_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_1523_power__less__power__Suc,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1524_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_1525_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1526_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_1527_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(times_times_int(A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_1528_not__sum__squares__lt__zero,axiom,
    ! [Xd: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_1529_not__sum__squares__lt__zero,axiom,
    ! [Xd: 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,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_1530_not__sum__squares__lt__zero,axiom,
    ! [Xd: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_1531_not__sum__squares__lt__zero,axiom,
    ! [Xd: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_1532_sum__squares__gt__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y)))
    <=> ( ( Xd != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1533_sum__squares__gt__zero__iff,axiom,
    ! [Xd: 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,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xd != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1534_sum__squares__gt__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( Xd != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1535_sum__squares__gt__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y)))
    <=> ( ( Xd != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1536_power__gt1,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1537_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_1538_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_1539_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_1540_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_1541_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_1542_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_1543_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_1544_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_1545_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_1546_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_1547_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_1548_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_1549_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_1550_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_1551_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_1552_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_1553_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xd: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_1554_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xd: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_1555_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_1556_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xd)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(Xd,Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_1557_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_1558_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_1559_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1560_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1561_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_1562_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(B),A)),B) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_1563_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_1564_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),B) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_1565_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_1566_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(times_times_int(A),C3) ) ) ).

% unit_dvdE
tff(fact_1567_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C3) ) ) ).

% unit_dvdE
tff(fact_1568_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1569_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1570_one__less__mult,axiom,
    ! [Nb: nat,M: 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)),M)
       => 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,M),Nb)) ) ) ).

% one_less_mult
tff(fact_1571_n__less__m__mult__n,axiom,
    ! [Nb: nat,M: 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)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1572_n__less__n__mult__m,axiom,
    ! [Nb: nat,M: 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)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1573_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_1574_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_1575_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_1576_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_1577_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_1578_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_eq
tff(fact_1579_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_1580_mod__add__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_1581_mod__add__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A3),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_1582_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B5),C) ) ) ) ).

% mod_add_cong
tff(fact_1583_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_1584_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_left_eq
tff(fact_1585_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_1586_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_1587_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),B)),C) ).

% mod_add_right_eq
tff(fact_1588_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_1589_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_1590_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_eq
tff(fact_1591_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_eq
tff(fact_1592_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1593_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A3),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1594_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A3),B5),C) ) ) ) ).

% mod_mult_cong
tff(fact_1595_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_1596_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_1597_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_1598_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_1599_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(times_times_int(C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ).

% mult_mod_right
tff(fact_1600_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ).

% mult_mod_right
tff(fact_1601_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1602_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_left_eq
tff(fact_1603_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_left_eq
tff(fact_1604_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1605_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),B)),C) ).

% mod_mult_right_eq
tff(fact_1606_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ).

% mod_mult_right_eq
tff(fact_1607_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1608_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: 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,M),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1609_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_1610_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),Nb)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Nb)),B) ).

% power_mod
tff(fact_1611_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),B) ).

% power_mod
tff(fact_1612_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1613_dvd__mult__cancel,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1614_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1615_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1616_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1617_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_1618_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_1619_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_1620_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_1621_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_1622_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_1623_dvd__mod,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( 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,M),Nb)) ) ) ).

% dvd_mod
tff(fact_1624_dvd__mod,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(M),Nb)) ) ) ).

% dvd_mod
tff(fact_1625_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( 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(M,Nb)) ) ) ).

% dvd_mod
tff(fact_1626_mod__Suc__eq,axiom,
    ! [M: 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,M),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% mod_Suc_eq
tff(fact_1627_mod__Suc__Suc__eq,axiom,
    ! [M: 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,M),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_1628_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) )
       => ~ ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
             => ( K != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_1629_Suc__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_lessD
tff(fact_1630_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
           => ( K != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_1631_Suc__lessI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( ( aa_nat_nat(suc,M) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb) ) ) ).

% Suc_lessI
tff(fact_1632_less__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ( M = Nb ) ) ) ).

% less_SucE
tff(fact_1633_less__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_1634_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_1635_less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% less_Suc_eq
tff(fact_1636_not__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_1637_Nat_OAll__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) ) ) ) ).

% Nat.All_less_Suc
tff(fact_1638_Suc__less__eq2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M)
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M5) ) ) ).

% Suc_less_eq2
tff(fact_1639_less__antisym,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
       => ( M = Nb ) ) ) ).

% less_antisym
tff(fact_1640_Suc__less__SucD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_SucD
tff(fact_1641_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_1642_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,J3: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J3),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_1643_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_1644_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1645_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: 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,M),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,M),Nb) ) ) ).

% less_add_eq_less
tff(fact_1646_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: 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,M),J)) ) ).

% trans_less_add2
tff(fact_1647_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: 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),M)) ) ).

% trans_less_add1
tff(fact_1648_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_1649_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_1650_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_1651_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_1652_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_1653_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(M))),numeral_numeral_nat(bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(Nb))),numeral_numeral_nat(bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1654_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1655_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(Nb)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1656_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1657_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1658_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1659_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_1660_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != numeral_numeral_real(Nb) ).

% zero_neq_numeral
tff(fact_1661_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != numeral_numeral_rat(Nb) ).

% zero_neq_numeral
tff(fact_1662_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != numeral_numeral_nat(Nb) ).

% zero_neq_numeral
tff(fact_1663_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_1664_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1665_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1666_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1667_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1668_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_1669_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1670_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1671_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1672_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1673_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1674_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_1675_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1676_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1677_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1678_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1679_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(times_times_int(A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1680_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_1681_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1682_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1683_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1684_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1685_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(C),A) = aa_int_int(times_times_int(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1686_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1687_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1688_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1689_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1690_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1691_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(times_times_int(A),C) = aa_int_int(times_times_int(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1692_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1693_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1694_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_1695_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_1696_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_1697_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_1698_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_1699_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_1700_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_1701_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_1702_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_1703_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_1704_one__dvd,axiom,
    ! [A: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,one_one_assn),A) ).

% one_dvd
tff(fact_1705_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_1706_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_1707_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_1708_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_1709_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1710_semiring__1__no__zero__divisors__class_Opower__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 ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1711_semiring__1__no__zero__divisors__class_Opower__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 ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1712_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1713_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1714_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1715_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1716_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B)
       => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1717_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1718_list__decode_Ocases,axiom,
    ! [Xd: nat] :
      ( ( Xd != zero_zero_nat )
     => ~ ! [N: nat] : Xd != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1719_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_1720_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1721_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1722_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1723_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1724_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_1725_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X),zero_zero_nat)
     => ( ! [Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y2))
       => ( ! [X: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X)),aa_nat_nat(suc,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_1726_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_1727_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1728_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1729_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1730_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_1731_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_1732_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_1733_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_1734_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_1735_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_1736_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_1737_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_1738_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_1739_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_1740_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_1741_add__eq__self__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = M )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1742_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
       => ( ! [A4: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_1743_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_1744_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(one_one_assn),P) = P ).

% assn_one_left
tff(fact_1745_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1746_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_1747_nat__mod__eq__iff,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),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,Xd),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_1748_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1749_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1750_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_1751_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_1752_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_1753_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1754_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1755_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1756_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1757_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(bit0(one))) ).

% pos2
tff(fact_1758_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 )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B2 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B2) = A )
                   => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B2) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1759_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 )
         => ! [B2: nat] :
              ( ( B2 != 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,one_one_nat),A) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = 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),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1760_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 )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B2 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B2) = A )
                   => ( ( aa_int_int(times_times_int(A),B2) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(times_times_int(C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1761_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1762_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1763_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1764_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1765_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1766_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1767_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J2) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_1768_dividend__less__div__times,axiom,
    ! [Nb: nat,M: 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,M),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,M),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1769_dividend__less__times__div,axiom,
    ! [Nb: nat,M: 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,M),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,M),Nb)))) ) ).

% dividend_less_times_div
tff(fact_1770_zero__less__power__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq
tff(fact_1771_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,numeral_numeral_nat(bit0(one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_1772_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,numeral_numeral_nat(bit0(one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_1773_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,numeral_numeral_nat(bit0(one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_1774_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1775_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,bit0(one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1776_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1777_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1778_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,bit0(one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1779_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1780_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(M))),numeral_numeral_nat(bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(one)),numeral_numeral_nat(bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1781_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1782_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1783_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(one)),numeral_numeral_nat(bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(Nb))),numeral_numeral_nat(bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1784_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1785_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(Nb)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1786_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D2: int] : B != aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(C),D2)) ) ).

% mod_eqE
tff(fact_1787_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),D2)) ) ).

% mod_eqE
tff(fact_1788_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_1789_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_1790_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_1791_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(bit0(one)))) ) ).

% half_gt_zero
tff(fact_1792_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(bit0(one)))) ) ).

% half_gt_zero
tff(fact_1793_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(bit0(one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_1794_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(bit0(one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_1795_power2__less__0,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_1796_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),numeral_numeral_nat(bit0(one)))),zero_zero_real) ).

% power2_less_0
tff(fact_1797_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_1798_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one)))),zero_zero_int) ).

% power2_less_0
tff(fact_1799_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),numeral_numeral_nat(bit0(one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1800_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),numeral_numeral_nat(bit0(one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1801_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_1802_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),numeral_numeral_nat(bit0(one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_1803_div__mod__decomp,axiom,
    ! [A2: nat,Nb: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_1804_one__plus__numeral__commute,axiom,
    ! [Xd: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xd)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xd)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_1805_one__plus__numeral__commute,axiom,
    ! [Xd: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Xd)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Xd)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1806_one__plus__numeral__commute,axiom,
    ! [Xd: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(Xd)) = plus_plus_rat(numeral_numeral_rat(Xd),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1807_one__plus__numeral__commute,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(Xd)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Xd)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1808_one__plus__numeral__commute,axiom,
    ! [Xd: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Xd)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Xd)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1809_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1810_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_1811_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_1812_numeral__One,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numeral_One
tff(fact_1813_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1814_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1815_less__iff__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1816_less__add__Suc2,axiom,
    ! [I: nat,M: 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,M),I))) ).

% less_add_Suc2
tff(fact_1817_less__add__Suc1,axiom,
    ! [I: nat,M: 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),M))) ).

% less_add_Suc1
tff(fact_1818_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ~ ! [Q4: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ) ).

% less_natE
tff(fact_1819_left__right__inverse__power,axiom,
    ! [Xd: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xd),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xd),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_1820_left__right__inverse__power,axiom,
    ! [Xd: assn,Y: assn,Nb: nat] :
      ( ( aa_assn_assn(times_times_assn(Xd),Y) = one_one_assn )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(Xd),Nb)),aa_nat_assn(power_power_assn(Y),Nb)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_1821_left__right__inverse__power,axiom,
    ! [Xd: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_1822_left__right__inverse__power,axiom,
    ! [Xd: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xd),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,Xd),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_1823_left__right__inverse__power,axiom,
    ! [Xd: int,Y: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xd),Y) = one_one_int )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xd),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_1824_left__right__inverse__power,axiom,
    ! [Xd: code_integer,Y: code_integer,Nb: nat] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_1825_left__right__inverse__power,axiom,
    ! [Xd: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xd),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xd),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_1826_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: 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)),M)),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,M),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1827_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_1828_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_1829_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),Nb) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_1830_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1831_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(times_times_int(B),A) = aa_int_int(times_times_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1832_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_1833_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1834_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1835_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_1836_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1837_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1838_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1839_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1840_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1841_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1842_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1843_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1844_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_1845_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1846_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1847_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_1848_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_1849_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_1850_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_1851_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_1852_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_1853_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_1854_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_1855_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_1856_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_1857_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xd: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xd),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xd),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1858_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xd: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1859_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xd: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(Xd,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(Xd),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_1860_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_1861_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_1862_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_1863_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_1864_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_1865_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_1866_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1867_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1868_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1869_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1870_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1871_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_1872_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1873_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1874_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1875_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1876_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1877_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1878_numerals_I1_J,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numerals(1)
tff(fact_1879_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),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,M),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_1880_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_1881_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1882_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_1883_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1884_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1885_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1886_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1887_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1888_one__is__add,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1889_add__is__1,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1890_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_1891_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))) = A )
     => ( aa_int_int(plus_plus_int(A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_1892_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_1893_pow_Osimps_I1_J,axiom,
    ! [Xd: num] : aa_num_num(aa_num_fun_num_num(pow,Xd),one) = Xd ).

% pow.simps(1)
tff(fact_1894_not__sum__power2__lt__zero,axiom,
    ! [Xd: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_1895_not__sum__power2__lt__zero,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_1896_not__sum__power2__lt__zero,axiom,
    ! [Xd: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_1897_not__sum__power2__lt__zero,axiom,
    ! [Xd: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_1898_sum__power2__gt__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one)))))
    <=> ( ( Xd != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1899_sum__power2__gt__zero__iff,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))
    <=> ( ( Xd != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1900_sum__power2__gt__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one)))))
    <=> ( ( Xd != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1901_sum__power2__gt__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one)))))
    <=> ( ( Xd != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1902_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),numeral_numeral_nat(bit0(one)))) ) ).

% div_2_gt_zero
tff(fact_1903_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)),numeral_numeral_nat(bit0(one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1904_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1905_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_1906_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_1907_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1908_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1909_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_1910_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1911_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(plus_plus_int(A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1912_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_1913_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_1914_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_1915_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_1916_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_1917_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_1918_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_1919_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1920_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1921_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1922_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1923_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),B)),aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))) = A ).

% mod_mult_div_eq
tff(fact_1924_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_1925_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_1926_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_1927_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_1928_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_1929_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_1930_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_1931_odd__power__less__zero,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_1932_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,numeral_numeral_nat(bit0(one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_1933_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,numeral_numeral_nat(bit0(one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_1934_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,numeral_numeral_nat(bit0(one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_1935_mod__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% mod_mult2_eq
tff(fact_1936_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1937_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1938_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1939_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_1940_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1941_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1942_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1943_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_1944_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1945_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : plus_plus_rat(B,plus_plus_rat(A,C)) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1946_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1947_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(plus_plus_int(B),aa_int_int(plus_plus_int(A),C)) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1948_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_1949_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,B) = plus_plus_rat(B,A) ).

% ab_semigroup_add_class.add.commute
tff(fact_1950_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_1951_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_1952_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1953_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1954_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A) = aa_int_int(plus_plus_int(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1955_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1956_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1957_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1958_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_1959_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).

% add.assoc
tff(fact_1960_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_1961_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),B)),C) = aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% add.assoc
tff(fact_1962_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( 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),B)) ) ) ).

% group_cancel.add2
tff(fact_1963_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = plus_plus_rat(K,B) )
     => ( plus_plus_rat(A,B3) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add2
tff(fact_1964_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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),B)) ) ) ).

% group_cancel.add2
tff(fact_1965_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(plus_plus_int(A),B3) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add2
tff(fact_1966_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1967_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( plus_plus_rat(A2,B) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).

% group_cancel.add1
tff(fact_1968_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1969_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A),B)) ) ) ).

% group_cancel.add1
tff(fact_1970_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_1971_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( plus_plus_rat(I,K) = plus_plus_rat(J,L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1972_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_1973_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(plus_plus_int(I),K) = aa_int_int(plus_plus_int(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1974_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C) = aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) ).

% mult.assoc
tff(fact_1975_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_1976_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_1977_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A),B)),C) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% mult.assoc
tff(fact_1978_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% mult.assoc
tff(fact_1979_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),B)),C) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% mult.assoc
tff(fact_1980_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(times_times_assn(A),B) = aa_assn_assn(times_times_assn(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1981_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1982_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1983_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),B) = aa_int_int(times_times_int(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1984_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1985_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),B) = aa_complex_complex(times_times_complex(B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1986_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(times_times_assn(B),aa_assn_assn(times_times_assn(A),C)) = aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1987_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1988_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1989_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(times_times_int(B),aa_int_int(times_times_int(A),C)) = aa_int_int(times_times_int(A),aa_int_int(times_times_int(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1990_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1991_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(times_times_complex(A),C)) = aa_complex_complex(times_times_complex(A),aa_complex_complex(times_times_complex(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1992_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1993_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1994_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = C )
      <=> ( A = aa_int_int(times_times_int(C),B) ) ) ) ).

% unit_eq_div1
tff(fact_1995_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1996_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1997_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(divide_divide_int(C),B) )
      <=> ( aa_int_int(times_times_int(A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_1998_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_1999_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2000_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2001_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),B) ) ) ).

% unit_div_commute
tff(fact_2002_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_2003_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),C)),B) ) ) ).

% unit_div_commute
tff(fact_2004_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_2005_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2006_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(times_times_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2007_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2008_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2009_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2010_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2011_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C),numeral_numeral_real(W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2012_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2013_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2014_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B,numeral_numeral_real(W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2015_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2016_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2017_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2018_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      plus_plus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(plus_plus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2019_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2020_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2021_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      plus_plus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2022_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xd: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xd),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2023_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xd: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2024_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xd: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(Xd,Y),divide_divide_rat(W,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2025_add__frac__num,axiom,
    ! [Y: complex,Xd: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xd),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2026_add__frac__num,axiom,
    ! [Y: real,Xd: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2027_add__frac__num,axiom,
    ! [Y: rat,Xd: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(Xd,Y),Z) = divide_divide_rat(plus_plus_rat(Xd,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2028_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xd: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xd),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xd),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2029_add__num__frac,axiom,
    ! [Y: real,Z: real,Xd: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2030_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xd: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(Z,divide_divide_rat(Xd,Y)) = divide_divide_rat(plus_plus_rat(Xd,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2031_add__divide__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xd),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xd),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2032_add__divide__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2033_add__divide__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(Xd,divide_divide_rat(Y,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Z),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2034_divide__add__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xd),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2035_divide__add__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2036_divide__add__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(Xd,Z),Y) = divide_divide_rat(plus_plus_rat(Xd,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2037_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2038_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2039_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(times_times_int(C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2040_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2041_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2042_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(times_times_int(C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2043_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2044_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2045_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2046_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2047_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2048_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(times_times_int(B),C) = aa_int_int(times_times_int(A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2049_numeral__1__eq__Suc__0,axiom,
    numeral_numeral_nat(one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_2050_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2051_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),one_one_nat) ).

% odd_one
tff(fact_2052_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% odd_one
tff(fact_2053_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),numeral_numeral_nat(bit0(one))) = one_one_rat ).

% one_power2
tff(fact_2054_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% one_power2
tff(fact_2055_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),numeral_numeral_nat(bit0(one))) = one_one_real ).

% one_power2
tff(fact_2056_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),numeral_numeral_nat(bit0(one))) = one_one_int ).

% one_power2
tff(fact_2057_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),numeral_numeral_nat(bit0(one))) = one_one_complex ).

% one_power2
tff(fact_2058_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),numeral_numeral_nat(bit0(one))) = one_one_Code_integer ).

% one_power2
tff(fact_2059_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),zero_zero_nat) ).

% even_zero
tff(fact_2060_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),zero_zero_int) ).

% even_zero
tff(fact_2061_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),numeral_numeral_nat(bit0(one))) = zero_zero_rat ).

% zero_power2
tff(fact_2062_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% zero_power2
tff(fact_2063_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),numeral_numeral_nat(bit0(one))) = zero_zero_real ).

% zero_power2
tff(fact_2064_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),numeral_numeral_nat(bit0(one))) = zero_zero_int ).

% zero_power2
tff(fact_2065_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),numeral_numeral_nat(bit0(one))) = zero_zero_complex ).

% zero_power2
tff(fact_2066_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),numeral_numeral_nat(bit0(one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_2067_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% nat_1_add_1
tff(fact_2068_numeral__2__eq__2,axiom,
    numeral_numeral_nat(bit0(one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_2069_field__less__half__sum,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),numeral_numeral_real(bit0(one)))) ) ).

% field_less_half_sum
tff(fact_2070_field__less__half__sum,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),divide_divide_rat(plus_plus_rat(Xd,Y),numeral_numeral_rat(bit0(one)))) ) ).

% field_less_half_sum
tff(fact_2071_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2072_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2073_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2074_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2075_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2076_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2077_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2078_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2079_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2080_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
     => ~ ! [B2: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B2),one_one_Code_integer) ) ).

% oddE
tff(fact_2081_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ~ ! [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,numeral_numeral_nat(bit0(one))),B2)),one_one_nat) ) ).

% oddE
tff(fact_2082_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
     => ~ ! [B2: int] : A != aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B2)),one_one_int) ) ).

% oddE
tff(fact_2083_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != bit0(X2) ).

% verit_eq_simplify(10)
tff(fact_2084_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( 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),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2085_nat__add__offset__less,axiom,
    ! [Y: nat,Nb: nat,Xd: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),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,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_2086_replicate__eq__replicate,axiom,
    ! [M: nat,Xd: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,Xd) = replicate_VEBT_VEBT(Nb,Y) )
    <=> ( ( M = Nb )
        & ( ( M != zero_zero_nat )
         => ( Xd = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_2087_nat__mod__eq_H,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = A ) ) ).

% nat_mod_eq'
tff(fact_2088_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q2: nat] :
      ( 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,R2),B)
       => 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,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q2),C))),R2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mod_lemma
tff(fact_2089_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(bit0(one))))) ) ).

% even_even_mod_4_iff
tff(fact_2090_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: 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(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),K) ) ) ).

% msrevs(1)
tff(fact_2091_n__less__equal__power__2,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,numeral_numeral_nat(bit0(one))),Nb)) ).

% n_less_equal_power_2
tff(fact_2092_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% set_bit_0
tff(fact_2093_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% set_bit_0
tff(fact_2094_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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))) ).

% set_bit_0
tff(fact_2095_not__real__square__gt__zero,axiom,
    ! [Xd: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd))
    <=> ( Xd = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_2096_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( ( aa_int_int(divide_divide_int(A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_2097_semiring__norm_I78_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(bit0(M)),bit0(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(78)
tff(fact_2098_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_num(M),one) ).

% semiring_norm(75)
tff(fact_2099_one__mod__exp__eq__one,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_2100_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_2101_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_num(one),bit0(Nb)) ).

% semiring_norm(76)
tff(fact_2102_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,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_2103_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(modulo_modulo_int(A2),Nb) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B3),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A2),Nb)),aa_int_int(divide_divide_int(B3),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2104_real__arch__pow__inv,axiom,
    ! [Y: real,Xd: 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,Xd),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_2105_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_2106_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_2107_int__div__less__self,axiom,
    ! [Xd: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(Xd),K)),Xd) ) ) ).

% int_div_less_self
tff(fact_2108_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_2109_zdiv__mult__self,axiom,
    ! [M: int,A: int,Nb: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(M),Nb))),M) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(A),M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_2110_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_2111_mod__exp__less__eq__exp,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% mod_exp_less_eq_exp
tff(fact_2112_pos__mod__bound2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% pos_mod_bound2
tff(fact_2113_axxmod2,axiom,
    ! [Xd: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Xd)),Xd)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int )
      & ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),Xd)),Xd)),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_2114_nmod2,axiom,
    ! [Nb: int] :
      ( ( aa_int_int(modulo_modulo_int(Nb),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int )
      | ( aa_int_int(modulo_modulo_int(Nb),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% nmod2
tff(fact_2115_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_2116_div__mod__decomp__int,axiom,
    ! [A2: int,Nb: int] : A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(divide_divide_int(A2),Nb)),Nb)),aa_int_int(modulo_modulo_int(A2),Nb)) ).

% div_mod_decomp_int
tff(fact_2117_real__arch__pow,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),N)) ) ).

% real_arch_pow
tff(fact_2118_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_2119_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)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb) = A )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb) = A ) )
               => ( Y3 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2120_p1mod22k_H,axiom,
    ! [B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))) ).

% p1mod22k'
tff(fact_2121_p1mod22k,axiom,
    ! [B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))),one_one_int) ).

% p1mod22k
tff(fact_2122_z1pmod2,axiom,
    ! [B: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% z1pmod2
tff(fact_2123_axxdiv2,axiom,
    ! [Xd: int] :
      ( ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Xd)),Xd)),aa_num_int(numeral_numeral_int,bit0(one))) = Xd )
      & ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),Xd)),Xd)),aa_num_int(numeral_numeral_int,bit0(one))) = Xd ) ) ).

% axxdiv2
tff(fact_2124_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,T),X4) ) ).

% minf(7)
tff(fact_2125_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X4) ) ).

% minf(7)
tff(fact_2126_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => ~ aa_num_o(ord_less_num(T),X4) ) ).

% minf(7)
tff(fact_2127_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X4) ) ).

% minf(7)
tff(fact_2128_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4) ) ).

% minf(7)
tff(fact_2129_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),T) ) ).

% minf(5)
tff(fact_2130_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),T) ) ).

% minf(5)
tff(fact_2131_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => aa_num_o(ord_less_num(X4),T) ) ).

% minf(5)
tff(fact_2132_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),T) ) ).

% minf(5)
tff(fact_2133_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T) ) ).

% minf(5)
tff(fact_2134_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_2135_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_2136_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_2137_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_2138_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( X4 != T ) ) ).

% minf(4)
tff(fact_2139_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_2140_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_2141_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_2142_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_2143_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( X4 != T ) ) ).

% minf(3)
tff(fact_2144_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q3,X) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                | aa_real_o(Q3,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_2145_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q3,X) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                | aa_rat_o(Q3,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_2146_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X: num] :
          ( aa_num_o(ord_less_num(X),Z3)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(ord_less_num(X),Z3)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q3,X) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(ord_less_num(X4),Z2)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                | aa_num_o(Q3,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_2147_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q3,X) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                | aa_nat_o(Q3,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_2148_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q3,X) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                | aa_int_o(Q3,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_2149_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q3,X) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                & aa_real_o(Q3,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_2150_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q3,X) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                & aa_rat_o(Q3,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_2151_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X: num] :
          ( aa_num_o(ord_less_num(X),Z3)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(ord_less_num(X),Z3)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q3,X) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(ord_less_num(X4),Z2)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                & aa_num_o(Q3,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_2152_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q3,X) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                & aa_nat_o(Q3,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_2153_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q3,X) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                & aa_int_o(Q3,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_2154_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,T),X4) ) ).

% pinf(7)
tff(fact_2155_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X4) ) ).

% pinf(7)
tff(fact_2156_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => aa_num_o(ord_less_num(T),X4) ) ).

% pinf(7)
tff(fact_2157_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X4) ) ).

% pinf(7)
tff(fact_2158_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4) ) ).

% pinf(7)
tff(fact_2159_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X4),T) ) ).

% pinf(5)
tff(fact_2160_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),T) ) ).

% pinf(5)
tff(fact_2161_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => ~ aa_num_o(ord_less_num(X4),T) ) ).

% pinf(5)
tff(fact_2162_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),T) ) ).

% pinf(5)
tff(fact_2163_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T) ) ).

% pinf(5)
tff(fact_2164_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_2165_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_2166_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_2167_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_2168_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(4)
tff(fact_2169_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_2170_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_2171_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_2172_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_2173_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( X4 != T ) ) ).

% pinf(3)
tff(fact_2174_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q3,X) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                | aa_real_o(Q3,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_2175_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q3,X) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                | aa_rat_o(Q3,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_2176_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X: num] :
          ( aa_num_o(ord_less_num(Z3),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(ord_less_num(Z3),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q3,X) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(ord_less_num(Z2),X4)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                | aa_num_o(Q3,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_2177_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q3,X) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                | aa_nat_o(Q3,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_2178_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q3,X) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                | aa_int_o(Q3,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_2179_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q3: fun_real_o] :
      ( ? [Z3: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q3,X) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                & aa_real_o(Q3,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_2180_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q3: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q3,X) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                & aa_rat_o(Q3,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_2181_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q3: fun_num_o] :
      ( ? [Z3: num] :
        ! [X: num] :
          ( aa_num_o(ord_less_num(Z3),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(ord_less_num(Z3),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q3,X) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(ord_less_num(Z2),X4)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                & aa_num_o(Q3,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_2182_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q3: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q3,X) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                & aa_nat_o(Q3,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_2183_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q3: fun_int_o] :
      ( ? [Z3: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q3,X) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                & aa_int_o(Q3,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_2184_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xd: nat] :
      ( ( Xd != zero_zero_nat )
     => ( ( Xd != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : Xd != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_2185_nat__mod__eq,axiom,
    ! [B: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Nb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = B ) ) ) ).

% nat_mod_eq
tff(fact_2186_mod__plus__right,axiom,
    ! [A: nat,Xd: nat,M: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Xd)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Xd)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),M) ) ) ).

% mod_plus_right
tff(fact_2187_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_2188_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),bit_se2793503036327961859nteger(M,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_2189_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_2190_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),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_2191_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S)) ) ) ).

% pinf(9)
tff(fact_2192_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),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_2193_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)) ) ) ).

% pinf(9)
tff(fact_2194_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),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_2195_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S)) ) ) ).

% pinf(10)
tff(fact_2196_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),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_2197_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)) ) ) ).

% pinf(10)
tff(fact_2198_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( 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_2199_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S)) ) ) ).

% minf(9)
tff(fact_2200_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( 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_2201_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)) ) ) ).

% minf(9)
tff(fact_2202_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( ~ 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_2203_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,S)) ) ) ).

% minf(10)
tff(fact_2204_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( ~ 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_2205_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),S)) ) ) ).

% minf(10)
tff(fact_2206_nat__mod__lem,axiom,
    ! [Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Nb)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) = B ) ) ) ).

% nat_mod_lem
tff(fact_2207_msrevs_I2_J,axiom,
    ! [K: nat,Nb: nat,M: 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,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) ).

% msrevs(2)
tff(fact_2208_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(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% set_bit_Suc
tff(fact_2209_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% set_bit_Suc
tff(fact_2210_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),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))))) ).

% set_bit_Suc
tff(fact_2211_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X5: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X5))
    <=> ? [X5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),plus_plus_rat(X5,zero_zero_rat))
          & aa_rat_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2212_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X5: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X5))
    <=> ? [X5: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X5),zero_zero_real))
          & aa_real_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2213_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X5: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X5))
    <=> ? [X5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),zero_zero_nat))
          & aa_nat_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2214_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X5: int] : aa_int_o(P,aa_int_int(times_times_int(L),X5))
    <=> ? [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(plus_plus_int(X5),zero_zero_int))
          & aa_int_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2215_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X5: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(L),X5))
    <=> ? [X5: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X5,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2216_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X5: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X5))
    <=> ? [X5: complex] :
          ( aa_complex_o(dvd_dvd_complex(L),aa_complex_complex(plus_plus_complex(X5),zero_zero_complex))
          & aa_complex_o(P,X5) ) ) ).

% unity_coeff_ex
tff(fact_2217_z1pdiv2,axiom,
    ! [B: int] : aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = B ).

% z1pdiv2
tff(fact_2218_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),B) ) ) ).

% td_gal_lt
tff(fact_2219_enat__ord__number_I2_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb)) ) ).

% enat_ord_number(2)
tff(fact_2220_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_2221_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat(A,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_2222_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(plus_plus_int(A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_2223_low__inv,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb))),Xd)),Nb) = Xd ) ) ).

% low_inv
tff(fact_2224_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_2225_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% flip_bit_Suc
tff(fact_2226_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),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_2227_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% unset_bit_0
tff(fact_2228_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% unset_bit_0
tff(fact_2229_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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))) ).

% unset_bit_0
tff(fact_2230_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_2231_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(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% unset_bit_Suc
tff(fact_2232_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),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_2233_low__def,axiom,
    ! [Xd: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% low_def
tff(fact_2234_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),Nb)),L) = Nb ) ) ) ).

% word_rot_lem
tff(fact_2235_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(bit0(one)) ).

% dbl_simps(3)
tff(fact_2236_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(bit0(one)) ).

% dbl_simps(3)
tff(fact_2237_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% dbl_simps(3)
tff(fact_2238_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% dbl_simps(3)
tff(fact_2239_mlex__snd__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,B5: nat,N2: nat] :
      ( ( A = A3 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B5)
       => 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,A),N2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N2)),B5)) ) ) ).

% mlex_snd_decrI
tff(fact_2240_bit__split__inv,axiom,
    ! [Xd: 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,Xd),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),D)),D) = Xd ).

% bit_split_inv
tff(fact_2241_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_2242_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_2243_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_2244_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_2245_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_2246_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_2247_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_2248_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_2249_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(bit0(K)) ).

% dbl_simps(5)
tff(fact_2250_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(bit0(K)) ).

% dbl_simps(5)
tff(fact_2251_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bit0(K)) ).

% dbl_simps(5)
tff(fact_2252_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,bit0(K)) ).

% dbl_simps(5)
tff(fact_2253_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(times_times_int(zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_2254_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(times_times_int(K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_2255_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q4: int] : M = aa_int_int(times_times_int(D),Q4) ) ).

% zmod_eq_0D
tff(fact_2256_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_2257_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),T)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(K),M)),aa_int_int(times_times_int(K),T)) ) ) ).

% zdvd_mono
tff(fact_2258_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q5: int] : M = aa_int_int(times_times_int(D),Q5) ) ).

% zmod_eq_0_iff
tff(fact_2259_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(times_times_int(K),I)),aa_int_int(times_times_int(K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_2260_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(times_times_int(M),Nb) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2261_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Nb))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_2262_zmod__helper,axiom,
    ! [Nb: int,M: int,K: int,A: int] :
      ( ( aa_int_int(modulo_modulo_int(Nb),M) = K )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Nb),A)),M) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(K),A)),M) ) ) ).

% zmod_helper
tff(fact_2263_mod__plus__cong,axiom,
    ! [B: int,B5: int,Xd: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B = B5 )
     => ( ( aa_int_int(modulo_modulo_int(Xd),B5) = aa_int_int(modulo_modulo_int(X6),B5) )
       => ( ( aa_int_int(modulo_modulo_int(Y),B5) = aa_int_int(modulo_modulo_int(Y4),B5) )
         => ( ( aa_int_int(plus_plus_int(X6),Y4) = Z4 )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Xd),Y)),B) = aa_int_int(modulo_modulo_int(Z4),B5) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_2264_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_2265_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_2266_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(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(plus_plus_int(I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_2267_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2268_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(plus_plus_int(K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_2269_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(plus_plus_int(zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_2270_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).

% neg_mod_bound
tff(fact_2271_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2272_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M3),N)
             => aa_Extended_enat_o(P,M3) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Nb) ) ).

% enat_less_induct
tff(fact_2273_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_2274_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_2275_zdvd__not__zless,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),M) ) ) ).

% zdvd_not_zless
tff(fact_2276_dbl__def,axiom,
    ! [Xd: real] : aa_real_real(neg_numeral_dbl_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Xd) ).

% dbl_def
tff(fact_2277_dbl__def,axiom,
    ! [Xd: rat] : neg_numeral_dbl_rat(Xd) = plus_plus_rat(Xd,Xd) ).

% dbl_def
tff(fact_2278_dbl__def,axiom,
    ! [Xd: int] : aa_int_int(neg_numeral_dbl_int,Xd) = aa_int_int(plus_plus_int(Xd),Xd) ).

% dbl_def
tff(fact_2279_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( 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),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2280_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_2281_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_2282_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se2159334234014336723it_int(M),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_2283_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_2284_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N2)
       => 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,A),N2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N2)) ) ) ).

% mlex_bound
tff(fact_2285_mlex__fst__decrI,axiom,
    ! [A: nat,A3: nat,B: nat,N2: nat,B5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B5),N2)
         => 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,A),N2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N2)),B5)) ) ) ) ).

% mlex_fst_decrI
tff(fact_2286_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xd: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xd))
         => 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),Xd)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_2287_forall__finite_I2_J,axiom,
    ! [P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P,I2) )
    <=> aa_nat_o(P,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_2288_forall__finite_I3_J,axiom,
    ! [Xd: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(suc,Xd)))
         => 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),aa_nat_nat(suc,Xd))
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_2289_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(bit0(one))),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% flip_bit_0
tff(fact_2290_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% flip_bit_0
tff(fact_2291_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))) ).

% flip_bit_0
tff(fact_2292_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( plus_plus_rat(A,aa_rat_rat(times_times_rat(R2),C)) != plus_plus_rat(B,aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2293_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2294_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2295_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(plus_plus_int(A),aa_int_int(times_times_int(R2),C)) != aa_int_int(plus_plus_int(B),aa_int_int(times_times_int(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2296_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(R2),C)) != plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2297_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2298_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_2299_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_2300_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_2301_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_2302_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_2303_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_2304_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_2305_of__bool__eq_I1_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_2306_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_2307_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_2308_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_2309_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_2310_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_2311_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_2312_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_2313_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_2314_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_2315_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_real(zero_n3304061248610475627l_real,(P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_2316_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_2317_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_2318_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_2319_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_2320_of__bool__eq_I2_J,axiom,
    aa_o_real(zero_n3304061248610475627l_real,$true) = one_one_real ).

% of_bool_eq(2)
tff(fact_2321_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_2322_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_2323_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_2324_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_2325_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2326_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_2327_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_2328_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_2329_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_2330_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2331_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_2332_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_2333_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_2334_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_2335_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_2336_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_2337_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_2338_odd__of__bool__self,axiom,
    ! [P3: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((P3)))
    <=> (P3) ) ).

% odd_of_bool_self
tff(fact_2339_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_2340_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(divide_divide_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_2341_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_2342_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2343_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2344_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_2345_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2346_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2347_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_2348_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_2349_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,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_2350_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(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_2351_imult__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_2352_iadd__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2353_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_2354_of__bool__eq__iff,axiom,
    ! [P3: $o,Q2: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)) )
    <=> ( (P3)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2355_of__bool__eq__iff,axiom,
    ! [P3: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P3)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P3)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2356_of__bool__eq__iff,axiom,
    ! [P3: $o,Q2: $o] :
      ( ( zero_n356916108424825756nteger((P3)) = zero_n356916108424825756nteger((Q2)) )
    <=> ( (P3)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_2357_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_real(zero_n3304061248610475627l_real,
        ( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_o_real(zero_n3304061248610475627l_real,(P))),aa_o_real(zero_n3304061248610475627l_real,(Q))) ).

% of_bool_conj
tff(fact_2358_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n1201886186963655149omplex(( (P)
        & (Q) )) = aa_complex_complex(times_times_complex(zero_n1201886186963655149omplex((P))),zero_n1201886186963655149omplex((Q))) ).

% of_bool_conj
tff(fact_2359_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_2360_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_2361_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_2362_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P3)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_2363_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P3)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_2364_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P3)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_2365_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P3)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_2366_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P3)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_2367_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P3)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_2368_split__of__bool,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ( ( (P3)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P3)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_2369_split__of__bool,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,aa_o_real(zero_n3304061248610475627l_real,(P3)))
    <=> ( ( (P3)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P3)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_2370_split__of__bool,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ( ( (P3)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P3)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_2371_split__of__bool,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P3)))
    <=> ( ( (P3)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P3)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_2372_split__of__bool,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ( ( (P3)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_2373_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P3: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P3)))
    <=> ( ( (P3)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P3)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_2374_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n1201886186963655149omplex((P3)) = $ite((P3),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_2375_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_real(zero_n3304061248610475627l_real,(P3)) = $ite((P3),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_2376_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n2052037380579107095ol_rat((P3)) = $ite((P3),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_2377_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P3)) = $ite((P3),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_2378_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P3)) = $ite((P3),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_2379_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n356916108424825756nteger((P3)) = $ite((P3),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_2380_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2381_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,bit0(one))),A)) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2382_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% of_bool_odd_eq_mod_2
tff(fact_2383_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),numeral_numeral_nat(bit0(one))) = A4 )
         => aa_nat_o(P,A4) )
     => ( ! [A4: nat,B2: $o] :
            ( aa_nat_o(P,A4)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A4))),numeral_numeral_nat(bit0(one))) = A4 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A4))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_2384_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(divide_divide_int(A4),aa_num_int(numeral_numeral_int,bit0(one))) = A4 )
         => aa_int_o(P,A4) )
     => ( ! [A4: int,B2: $o] :
            ( aa_int_o(P,A4)
           => ( ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A4))),aa_num_int(numeral_numeral_int,bit0(one))) = A4 )
             => aa_int_o(P,aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A4))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_2385_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(bit0(one))) = A4 )
         => aa_Code_integer_o(P,A4) )
     => ( ! [A4: code_integer,B2: $o] :
            ( aa_Code_integer_o(P,A4)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),A4)),numera6620942414471956472nteger(bit0(one))) = A4 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),A4))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_2386_exp__mod__exp,axiom,
    ! [M: 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,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ).

% exp_mod_exp
tff(fact_2387_exp__mod__exp,axiom,
    ! [M: nat,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) ).

% exp_mod_exp
tff(fact_2388_exp__mod__exp,axiom,
    ! [M: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) ).

% exp_mod_exp
tff(fact_2389_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_2390_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_2391_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = plus_plus_rat(B,A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_2392_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_2393_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(plus_plus_int(B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_2394_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),D)) != plus_plus_rat(aa_rat_rat(times_times_rat(A),D),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2395_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2396_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2397_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) != aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),D)),aa_int_int(times_times_int(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2398_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),D),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2399_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),D)) != aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),D)),aa_complex_complex(times_times_complex(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2400_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xd: rat,Z: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(W),Y),aa_rat_rat(times_times_rat(Xd),Z)) = plus_plus_rat(aa_rat_rat(times_times_rat(W),Z),aa_rat_rat(times_times_rat(Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2401_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xd: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2402_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xd: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xd),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2403_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xd: int,Z: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Y)),aa_int_int(times_times_int(Xd),Z)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(W),Z)),aa_int_int(times_times_int(Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2404_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,Xd: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(W),Y),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Z)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(W),Z),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2405_crossproduct__eq,axiom,
    ! [W: complex,Y: complex,Xd: complex,Z: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W),Y)),aa_complex_complex(times_times_complex(Xd),Z)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(W),Z)),aa_complex_complex(times_times_complex(Xd),Y)) )
    <=> ( ( W = Xd )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2406_forall__finite_I1_J,axiom,
    ! [P: fun_nat_o,I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),zero_zero_nat)
     => aa_nat_o(P,I4) ) ).

% forall_finite(1)
tff(fact_2407_mult__less__iff1,axiom,
    ! [Z: code_integer,Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Z)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xd),Y) ) ) ).

% mult_less_iff1
tff(fact_2408_mult__less__iff1,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ).

% mult_less_iff1
tff(fact_2409_mult__less__iff1,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xd),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y) ) ) ).

% mult_less_iff1
tff(fact_2410_mult__less__iff1,axiom,
    ! [Z: int,Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(Xd),Z)),aa_int_int(times_times_int(Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Y) ) ) ).

% mult_less_iff1
tff(fact_2411_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2412_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2413_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)))
         => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2414_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),numeral_numeral_nat(W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2415_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(W))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2416_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),numeral_numeral_nat(W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2417_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),numeral_numeral_nat(W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2418_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_2419_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_2420_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(bit0(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_num(X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_2421_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_2422_arith__geo__mean,axiom,
    ! [U: real,Xd: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),numeral_numeral_nat(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
       => ( 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,Xd),Y)),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2423_arith__geo__mean,axiom,
    ! [U: rat,Xd: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),numeral_numeral_nat(bit0(one))) = aa_rat_rat(times_times_rat(Xd),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),divide_divide_rat(plus_plus_rat(Xd,Y),numeral_numeral_rat(bit0(one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2424_less__option__None,axiom,
    ! [Xd: option_nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xd),none_nat) ).

% less_option_None
tff(fact_2425_less__option__None,axiom,
    ! [Xd: option_num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,Xd),none_num) ).

% less_option_None
tff(fact_2426_min__in__set__def,axiom,
    ! [Xs: set_nat,Xd: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xd)
    <=> ( aa_set_nat_o(member_nat(Xd),Xs)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),X5) ) ) ) ).

% min_in_set_def
tff(fact_2427_max__in__set__def,axiom,
    ! [Xs: set_nat,Xd: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xd)
    <=> ( aa_set_nat_o(member_nat(Xd),Xs)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xd) ) ) ) ).

% max_in_set_def
tff(fact_2428_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(bit0(M)),bit0(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(71)
tff(fact_2429_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_eq_num(one),Nb) ).

% semiring_norm(68)
tff(fact_2430_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% Suc_le_mono
tff(fact_2431_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_2432_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_2433_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_2434_less__eq__option__None__code,axiom,
    ! [Xd: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xd) ).

% less_eq_option_None_code
tff(fact_2435_less__eq__option__None__code,axiom,
    ! [Xd: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xd) ).

% less_eq_option_None_code
tff(fact_2436_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_2437_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_2438_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_2439_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_2440_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% numeral_le_iff
tff(fact_2441_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_2442_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2443_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2444_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2445_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_2446_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2447_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2448_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2449_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(bit0(M)),one) ).

% semiring_norm(69)
tff(fact_2450_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_2451_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_2452_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_2453_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_2454_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_2455_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb)) ) ).

% enat_ord_number(1)
tff(fact_2456_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_2457_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),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_2458_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2459_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_2460_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_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_2461_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2462_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2463_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),plus_plus_rat(B,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2464_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2465_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2466_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2467_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),plus_plus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2468_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2469_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2470_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2471_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2472_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2473_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2474_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2475_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(B,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2476_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2477_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2478_one__le__mult__iff,axiom,
    ! [M: 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,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & 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_2479_mult__le__cancel2,axiom,
    ! [M: 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,M),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,M),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2480_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2481_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_2482_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_2483_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_2484_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_2485_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_2486_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),one_one_real)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2487_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),one_one_rat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2488_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Nb)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2489_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(ord_less_eq_num(Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2490_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_2491_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2492_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_2493_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2494_power__increasing__iff,axiom,
    ! [B: code_integer,Xd: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y) ) ) ).

% power_increasing_iff
tff(fact_2495_power__increasing__iff,axiom,
    ! [B: real,Xd: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y) ) ) ).

% power_increasing_iff
tff(fact_2496_power__increasing__iff,axiom,
    ! [B: rat,Xd: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xd)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y) ) ) ).

% power_increasing_iff
tff(fact_2497_power__increasing__iff,axiom,
    ! [B: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y) ) ) ).

% power_increasing_iff
tff(fact_2498_power__increasing__iff,axiom,
    ! [B: int,Xd: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xd)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y) ) ) ).

% power_increasing_iff
tff(fact_2499_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2500_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2501_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2502_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2503_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_2504_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2505_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2506_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2507_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2508_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2509_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2510_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2511_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2512_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2513_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2514_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2515_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2516_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2517_power__decreasing__iff,axiom,
    ! [B: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2518_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2519_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2520_power__decreasing__iff,axiom,
    ! [B: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2521_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2522_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2523_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2524_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2525_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2526_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,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_2527_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),numeral_numeral_nat(bit0(one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2528_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one)))),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_2529_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),numeral_numeral_nat(bit0(one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2530_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),numeral_numeral_nat(bit0(one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2531_power2__eq__iff__nonneg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))) )
        <=> ( Xd = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2532_power2__eq__iff__nonneg,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))) )
        <=> ( Xd = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2533_power2__eq__iff__nonneg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))) )
        <=> ( Xd = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2534_power2__eq__iff__nonneg,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xd)
     => ( 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,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(bit0(one))) )
        <=> ( Xd = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2535_power2__eq__iff__nonneg,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))) )
        <=> ( Xd = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2536_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_2537_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),numeral_numeral_nat(W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_2538_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2539_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),numeral_numeral_nat(W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_2540_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),numeral_numeral_nat(W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),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_2541_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2542_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2543_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_2544_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2545_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2546_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2547_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2548_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(ord_less_eq_num(aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(ord_less_eq_num(aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_2549_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2550_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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_2551_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_2552_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_2553_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(ord_less_eq_num(A),A) ).

% verit_comp_simplify1(2)
tff(fact_2554_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_2555_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_2556_verit__la__generic,axiom,
    ! [A: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xd)
      | ( A = Xd )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),A) ) ).

% verit_la_generic
tff(fact_2557_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_2558_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(ord_less_eq_num(A),B)
      | ~ aa_num_o(ord_less_eq_num(B),A) ) ).

% verit_la_disequality
tff(fact_2559_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_2560_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_2561_less__eq__option__None,axiom,
    ! [Xd: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xd) ).

% less_eq_option_None
tff(fact_2562_less__eq__option__None,axiom,
    ! [Xd: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xd) ).

% less_eq_option_None
tff(fact_2563_less__eq__option__None__is__None,axiom,
    ! [Xd: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xd),none_nat)
     => ( Xd = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_2564_less__eq__option__None__is__None,axiom,
    ! [Xd: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,Xd),none_num)
     => ( Xd = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_2565_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% subset_decode_imp_le
tff(fact_2566_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( 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),N2)) ) ) ).

% power_increasing
tff(fact_2567_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% power_increasing
tff(fact_2568_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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),N2)) ) ) ).

% power_increasing
tff(fact_2569_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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),N2)) ) ) ).

% power_increasing
tff(fact_2570_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),N2)) ) ) ).

% power_increasing
tff(fact_2571_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(K),L)) ).

% signed_take_bit_add
tff(fact_2572_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% signed_take_bit_mult
tff(fact_2573_zero__le,axiom,
    ! [Xd: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xd) ).

% zero_le
tff(fact_2574_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_2575_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_2576_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_2577_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_2578_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A3: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B5),A3)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_2579_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A3: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B5),A3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_2580_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A3: num] :
      ( ~ aa_num_o(ord_less_eq_num(B5),A3)
    <=> aa_num_o(ord_less_num(A3),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_2581_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A3: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B5),A3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_2582_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A3: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),A3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_2583_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),T) ) ).

% pinf(6)
tff(fact_2584_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),T) ) ).

% pinf(6)
tff(fact_2585_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => ~ aa_num_o(ord_less_eq_num(X4),T) ) ).

% pinf(6)
tff(fact_2586_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),T) ) ).

% pinf(6)
tff(fact_2587_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T) ) ).

% pinf(6)
tff(fact_2588_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X4) ) ).

% pinf(8)
tff(fact_2589_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X4) ) ).

% pinf(8)
tff(fact_2590_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(Z2),X4)
     => aa_num_o(ord_less_eq_num(T),X4) ) ).

% pinf(8)
tff(fact_2591_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X4) ) ).

% pinf(8)
tff(fact_2592_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4) ) ).

% pinf(8)
tff(fact_2593_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),T) ) ).

% minf(6)
tff(fact_2594_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),T) ) ).

% minf(6)
tff(fact_2595_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => aa_num_o(ord_less_eq_num(X4),T) ) ).

% minf(6)
tff(fact_2596_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),T) ) ).

% minf(6)
tff(fact_2597_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T) ) ).

% minf(6)
tff(fact_2598_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X4) ) ).

% minf(8)
tff(fact_2599_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X4) ) ).

% minf(8)
tff(fact_2600_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(ord_less_num(X4),Z2)
     => ~ aa_num_o(ord_less_eq_num(T),X4) ) ).

% minf(8)
tff(fact_2601_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X4) ) ).

% minf(8)
tff(fact_2602_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4) ) ).

% minf(8)
tff(fact_2603_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_2604_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2605_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2606_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2607_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_2608_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2609_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2610_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2611_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_2612_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_2613_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)) ) ).

% add_right_mono
tff(fact_2614_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_2615_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_right_mono
tff(fact_2616_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_2617_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_2618_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)) ) ).

% add_left_mono
tff(fact_2619_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_2620_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_left_mono
tff(fact_2621_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_2622_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D)) ) ) ).

% add_mono
tff(fact_2623_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_2624_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_mono
tff(fact_2625_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_2626_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2627_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_2628_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2629_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_2630_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2631_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_2632_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2633_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_2634_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2635_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_2636_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2637_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_2638_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_2639_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_2640_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_2641_le__num__One__iff,axiom,
    ! [Xd: num] :
      ( aa_num_o(ord_less_eq_num(Xd),one)
    <=> ( Xd = one ) ) ).

% le_num_One_iff
tff(fact_2642_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% Suc_leD
tff(fact_2643_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_2644_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_2645_Suc__le__D,axiom,
    ! [Nb: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M6)
     => ? [M2: nat] : M6 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_2646_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        | ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_2647_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_2648_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_2649_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M3)),N)
             => aa_nat_o(P,M3) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_2650_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_2651_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X),X)
       => ( ! [X: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,M),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2652_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_2653_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_2654_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_2655_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_2656_exists__leI,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ( ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),Nb)
           => ~ aa_nat_o(P,N4) )
       => aa_nat_o(P,Nb) )
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),Nb)
          & aa_nat_o(P,N5) ) ) ).

% exists_leI
tff(fact_2657_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_2658_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_imp_le_nat
tff(fact_2659_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_2660_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_2661_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ( M != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_2662_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J3)) )
     => ( 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_2663_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_2664_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: 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,M),J)) ) ).

% trans_le_add2
tff(fact_2665_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: 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),M)) ) ).

% trans_le_add1
tff(fact_2666_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_2667_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_2668_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_2669_add__leD2,axiom,
    ! [M: 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,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_2670_add__leD1,axiom,
    ! [M: 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,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% add_leD1
tff(fact_2671_le__add2,axiom,
    ! [Nb: nat,M: 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,M),Nb)) ).

% le_add2
tff(fact_2672_le__add1,axiom,
    ! [Nb: nat,M: 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),M)) ).

% le_add1
tff(fact_2673_add__leE,axiom,
    ! [M: 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,M),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_2674_less__eq__real__def,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
        | ( Xd = Y ) ) ) ).

% less_eq_real_def
tff(fact_2675_conj__le__cong,axiom,
    ! [Xd: int,X6: int,P: $o,P2: $o] :
      ( ( Xd = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_2676_imp__le__cong,axiom,
    ! [Xd: int,X6: int,P: $o,P2: $o] :
      ( ( Xd = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_2677_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_2678_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_2679_div__le__dividend,axiom,
    ! [M: 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,M),Nb)),M) ).

% div_le_dividend
tff(fact_2680_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_2681_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_2682_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_2683_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_2684_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_2685_mod__less__eq__dividend,axiom,
    ! [M: 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,M),Nb)),M) ).

% mod_less_eq_dividend
tff(fact_2686_i0__lb,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Nb) ).

% i0_lb
tff(fact_2687_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_2688_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( 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),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2689_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2690_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,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),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2691_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2692_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),N2)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2693_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2694_power__le__imp__le__exp,axiom,
    ! [A: real,M: 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),M)),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,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2695_power__le__imp__le__exp,axiom,
    ! [A: rat,M: 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),M)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2696_power__le__imp__le__exp,axiom,
    ! [A: nat,M: 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),M)),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,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2697_power__le__imp__le__exp,axiom,
    ! [A: int,M: 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),M)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_2698_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_2699_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_2700_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Z)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2701_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2702_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xd),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2703_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xd),Z)),aa_int_int(times_times_int(Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2704_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),Xd)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),Y))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2705_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2706_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xd)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2707_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Z),Xd)),aa_int_int(times_times_int(Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2708_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_2709_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_2710_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_2711_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_2712_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_le_numeral
tff(fact_2713_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_le_numeral
tff(fact_2714_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),numeral_numeral_nat(Nb)) ).

% zero_le_numeral
tff(fact_2715_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_2716_add__nonpos__eq__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),Y) = zero_zero_real )
        <=> ( ( Xd = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2717_add__nonpos__eq__0__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( plus_plus_rat(Xd,Y) = zero_zero_rat )
        <=> ( ( Xd = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2718_add__nonpos__eq__0__iff,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),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,Xd),Y) = zero_zero_nat )
        <=> ( ( Xd = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2719_add__nonpos__eq__0__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(plus_plus_int(Xd),Y) = zero_zero_int )
        <=> ( ( Xd = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2720_add__nonneg__eq__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y) = zero_zero_real )
        <=> ( ( Xd = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2721_add__nonneg__eq__0__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( plus_plus_rat(Xd,Y) = zero_zero_rat )
        <=> ( ( Xd = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2722_add__nonneg__eq__0__iff,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xd)
     => ( 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,Xd),Y) = zero_zero_nat )
        <=> ( ( Xd = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2723_add__nonneg__eq__0__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(plus_plus_int(Xd),Y) = zero_zero_int )
        <=> ( ( Xd = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2724_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2725_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2726_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2727_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2728_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2729_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A,B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2730_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2731_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2732_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_2733_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),plus_plus_rat(A,C)) ) ) ).

% add_increasing2
tff(fact_2734_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_2735_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_increasing2
tff(fact_2736_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2737_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),B) ) ) ).

% add_decreasing2
tff(fact_2738_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2739_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2740_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_2741_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),plus_plus_rat(A,C)) ) ) ).

% add_increasing
tff(fact_2742_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_2743_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_increasing
tff(fact_2744_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2745_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,C)),B) ) ) ).

% add_decreasing
tff(fact_2746_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2747_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),C)),B) ) ) ).

% add_decreasing
tff(fact_2748_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_2749_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_2750_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_2751_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_2752_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_2753_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_2754_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_2755_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_2756_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_2757_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_2758_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_2759_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_2760_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2761_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2762_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2763_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2764_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2765_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2766_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2767_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2768_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2769_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2770_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_2771_zero__le__square,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A)) ).

% zero_le_square
tff(fact_2772_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_2773_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),A)) ).

% zero_le_square
tff(fact_2774_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2775_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ) ).

% split_mult_pos_le
tff(fact_2776_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_2777_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ).

% split_mult_pos_le
tff(fact_2778_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2779_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2780_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2781_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2782_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2783_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2784_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2785_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2786_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_2787_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% mult_left_mono
tff(fact_2788_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_2789_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2790_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% mult_left_mono
tff(fact_2791_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2792_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2793_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2794_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2795_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_2796_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ) ) ).

% mult_right_mono
tff(fact_2797_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_2798_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2799_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ) ) ).

% mult_right_mono
tff(fact_2800_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2801_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2802_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2803_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2804_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2805_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger) ) ).

% split_mult_neg_le
tff(fact_2806_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2807_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2808_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2809_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2810_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2811_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2812_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2813_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2814_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2815_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos
tff(fact_2816_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2817_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2818_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2819_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2820_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonpos_nonneg
tff(fact_2821_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2822_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2823_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2824_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2825_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2826_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2827_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2828_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2829_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2830_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_le_mult_iff
tff(fact_2831_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2832_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2833_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2834_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2835_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2836_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2837_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2838_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2839_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D)) ) ) ).

% add_less_le_mono
tff(fact_2840_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2841_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2842_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2843_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D)) ) ) ).

% add_le_less_mono
tff(fact_2844_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2845_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2846_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_2847_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2848_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_2849_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2850_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_2851_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,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2852_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_2853_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(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2854_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Nb)) ).

% one_le_numeral
tff(fact_2855_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(Nb)) ).

% one_le_numeral
tff(fact_2856_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(Nb)) ).

% one_le_numeral
tff(fact_2857_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_2858_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2859_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C)) ) ) ).

% divide_right_mono_neg
tff(fact_2860_divide__nonpos__nonpos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2861_divide__nonpos__nonpos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2862_divide__nonpos__nonneg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2863_divide__nonpos__nonneg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2864_divide__nonneg__nonpos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2865_divide__nonneg__nonpos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2866_divide__nonneg__nonneg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2867_divide__nonneg__nonneg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2868_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2869_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2870_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_2871_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_right_mono
tff(fact_2872_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2873_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2874_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_2875_zero__le__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% zero_le_power
tff(fact_2876_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_2877_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_2878_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_2879_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2880_power__mono,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono
tff(fact_2881_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2882_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2883_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_2884_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_2885_one__le__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% one_le_power
tff(fact_2886_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_2887_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_2888_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_2889_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2890_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2891_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2892_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct'
tff(fact_2893_nat__compl__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct
tff(fact_2894_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_2895_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_2896_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_2897_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_2898_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_Suc_eq_le
tff(fact_2899_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% le_less_Suc_eq
tff(fact_2900_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_lessD
tff(fact_2901_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_2902_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_2903_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_eq
tff(fact_2904_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb) ) ).

% Suc_leI
tff(fact_2905_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_2906_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_2907_nat__geq__1__eq__neqz,axiom,
    ! [Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Xd)
    <=> ( Xd != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2908_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2909_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2910_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2911_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2912_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2913_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2914_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2915_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2916_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2917_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B) ) ) ).

% power_le_dvd
tff(fact_2918_dvd__power__le,axiom,
    ! [Xd: nat,Y: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xd),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_2919_dvd__power__le,axiom,
    ! [Xd: real,Y: real,Nb: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M)) ) ) ).

% dvd_power_le
tff(fact_2920_dvd__power__le,axiom,
    ! [Xd: int,Y: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xd),Nb)),aa_nat_int(power_power_int(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2921_dvd__power__le,axiom,
    ! [Xd: complex,Y: complex,Nb: nat,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xd),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xd),Nb)),aa_nat_complex(power_power_complex(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2922_dvd__power__le,axiom,
    ! [Xd: code_integer,Y: code_integer,Nb: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_2923_Suc__div__le__mono,axiom,
    ! [M: 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,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Nb)) ).

% Suc_div_le_mono
tff(fact_2924_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: 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)),M)),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,M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_2925_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% zero_less_eq_of_bool
tff(fact_2926_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_2927_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_2928_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_2929_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_2930_mlex__leI,axiom,
    ! [A: nat,A3: nat,B: nat,B5: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B5)
       => 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,A),N2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N2)),B5)) ) ) ).

% mlex_leI
tff(fact_2931_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(zero_n3304061248610475627l_real,(P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_2932_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_2933_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_2934_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_2935_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_2936_mod__Suc__le__divisor,axiom,
    ! [M: 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,M),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2937_div__times__less__eq__dividend,axiom,
    ! [M: 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,M),Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_2938_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,M: 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,M),Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_2939_div__mult__le,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A) ).

% div_mult_le
tff(fact_2940_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(plus_plus_int(I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_2941_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2942_zdvd__antisym__nonneg,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),M)
           => ( M = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2943_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,bit0(one))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2944_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,bit0(one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2945_field__le__epsilon,axiom,
    ! [Xd: 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,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ).

% field_le_epsilon
tff(fact_2946_field__le__epsilon,axiom,
    ! [Xd: 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,Xd),plus_plus_rat(Y,E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y) ) ).

% field_le_epsilon
tff(fact_2947_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2948_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),plus_plus_rat(A,C)) ) ) ).

% add_strict_increasing2
tff(fact_2949_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2950_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2951_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2952_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),plus_plus_rat(A,C)) ) ) ).

% add_strict_increasing
tff(fact_2953_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2954_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A),C)) ) ) ).

% add_strict_increasing
tff(fact_2955_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2956_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B)) ) ) ).

% add_pos_nonneg
tff(fact_2957_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2958_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2959_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2960_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2961_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2962_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2963_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2964_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B)) ) ) ).

% add_nonneg_pos
tff(fact_2965_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2966_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2967_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2968_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2969_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2970_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2971_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2972_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2973_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2974_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2975_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2976_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2977_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2978_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2979_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2980_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2981_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2982_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2983_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2984_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2985_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2986_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2987_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2988_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2989_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2990_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2991_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2992_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2993_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2994_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2995_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2996_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2997_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2998_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2999_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3000_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3001_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3002_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3003_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3004_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3005_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3006_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3007_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3008_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3009_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3010_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3011_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3012_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3013_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3014_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3015_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3016_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3017_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3018_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3019_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),aa_int_int(times_times_int(C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3020_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3021_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3022_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3023_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3024_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3025_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3026_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3027_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3028_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3029_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3030_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3031_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3032_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3033_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3034_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3035_sum__squares__ge__zero,axiom,
    ! [Xd: 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,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3036_sum__squares__ge__zero,axiom,
    ! [Xd: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3037_sum__squares__ge__zero,axiom,
    ! [Xd: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3038_sum__squares__ge__zero,axiom,
    ! [Xd: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3039_sum__squares__le__zero__iff,axiom,
    ! [Xd: 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,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xd = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3040_sum__squares__le__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y))),zero_z3403309356797280102nteger)
    <=> ( ( Xd = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3041_sum__squares__le__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( Xd = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3042_sum__squares__le__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y))),zero_zero_int)
    <=> ( ( Xd = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3043_mult__left__le__one__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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),Xd)),Xd) ) ) ) ).

% mult_left_le_one_le
tff(fact_3044_mult__left__le__one__le,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Xd)),Xd) ) ) ) ).

% mult_left_le_one_le
tff(fact_3045_mult__left__le__one__le,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xd)),Xd) ) ) ) ).

% mult_left_le_one_le
tff(fact_3046_mult__left__le__one__le,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Y),Xd)),Xd) ) ) ) ).

% mult_left_le_one_le
tff(fact_3047_mult__right__le__one__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),Xd) ) ) ) ).

% mult_right_le_one_le
tff(fact_3048_mult__right__le__one__le,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Y)),Xd) ) ) ) ).

% mult_right_le_one_le
tff(fact_3049_mult__right__le__one__le,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xd),Y)),Xd) ) ) ) ).

% mult_right_le_one_le
tff(fact_3050_mult__right__le__one__le,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xd),Y)),Xd) ) ) ) ).

% mult_right_le_one_le
tff(fact_3051_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3052_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)),one_one_Code_integer) ) ) ) ).

% mult_le_one
tff(fact_3053_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3054_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3055_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3056_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_3057_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3058_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3059_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_3060_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3061_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_3062_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),one_one_int)),B) ) ).

% discrete
tff(fact_3063_frac__le,axiom,
    ! [Y: real,Xd: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3064_frac__le,axiom,
    ! [Y: rat,Xd: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_le
tff(fact_3065_frac__less,axiom,
    ! [Xd: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3066_frac__less,axiom,
    ! [Xd: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less
tff(fact_3067_frac__less2,axiom,
    ! [Xd: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3068_frac__less2,axiom,
    ! [Xd: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less2
tff(fact_3069_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3070_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3071_divide__nonneg__neg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3072_divide__nonneg__neg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3073_divide__nonneg__pos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3074_divide__nonneg__pos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3075_divide__nonpos__neg,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3076_divide__nonpos__neg,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xd,Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3077_divide__nonpos__pos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3078_divide__nonpos__pos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3079_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3080_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3081_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3082_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_positive
tff(fact_3083_power__less__imp__less__base,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3084_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3085_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3086_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3087_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3088_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3089_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3090_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3091_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_3092_power__le__one,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_3093_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_3094_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_3095_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_3096_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3097_power__inject__base,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,Nb)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3098_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3099_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3100_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3101_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3102_power__le__imp__le__base,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,Nb)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3103_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3104_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3105_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3106_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3107_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3108_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_3109_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3110_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3111_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_3112_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_3113_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_3114_div__le__mono2,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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),M)) ) ) ).

% div_le_mono2
tff(fact_3115_div__greater__zero__iff,axiom,
    ! [M: 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,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3116_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: 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),M)),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,M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3117_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_3118_mod__le__divisor,axiom,
    ! [Nb: nat,M: 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,M),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_3119_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_3120_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_3121_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_3122_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3123_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3124_int__mod__lem,axiom,
    ! [Nb: int,B: int] :
      ( 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,zero_zero_int),B)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),Nb) )
      <=> ( aa_int_int(modulo_modulo_int(B),Nb) = B ) ) ) ).

% int_mod_lem
tff(fact_3125_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_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_3126_int__mod__eq,axiom,
    ! [B: int,Nb: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Nb)
       => ( ( aa_int_int(modulo_modulo_int(A),Nb) = aa_int_int(modulo_modulo_int(B),Nb) )
         => ( aa_int_int(modulo_modulo_int(A),Nb) = B ) ) ) ) ).

% int_mod_eq
tff(fact_3127_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_3128_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% neg_mod_conj
tff(fact_3129_int__mod__ge,axiom,
    ! [A: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),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,A),aa_int_int(modulo_modulo_int(A),Nb)) ) ) ).

% int_mod_ge
tff(fact_3130_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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),M)
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_3131_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),Nb)
       => ~ ! [S2: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_3132_nat__mod__eq__lemma,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),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),Xd)
       => ? [Q4: nat] : Xd = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3133_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3134_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3135_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3136_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),A) ) ) ).

% zdiv_le_dividend
tff(fact_3137_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(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_3138_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3139_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3140_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)) ) ) ).

% div_positive_int
tff(fact_3141_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3142_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B5)),aa_int_int(divide_divide_int(A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3143_zdiv__mono1__neg,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A3),B)),aa_int_int(divide_divide_int(A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3144_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_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_3145_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_3146_zdiv__mono1,axiom,
    ! [A: int,A3: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A3),B)) ) ) ).

% zdiv_mono1
tff(fact_3147_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3148_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3149_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)) ) ) ) ).

% nonneg_mod_div
tff(fact_3150_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3151_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3152_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3153_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3154_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3155_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_3156_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3157_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3158_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3159_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3160_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3161_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3162_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3163_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_3164_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3165_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3166_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3167_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3168_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3169_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(times_times_int(C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3170_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3171_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_3172_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3173_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3174_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3175_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3176_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3177_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(times_times_int(B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3178_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3179_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_3180_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3181_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3182_field__le__mult__one__interval,axiom,
    ! [Xd: real,Y: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Xd)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ).

% field_le_mult_one_interval
tff(fact_3183_field__le__mult__one__interval,axiom,
    ! [Xd: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Xd)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y) ) ).

% field_le_mult_one_interval
tff(fact_3184_convex__bound__le,axiom,
    ! [Xd: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3185_convex__bound__le,axiom,
    ! [Xd: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( plus_p5714425477246183910nteger(U,V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(U),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3186_convex__bound__le,axiom,
    ! [Xd: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),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)
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),Xd),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3187_convex__bound__le,axiom,
    ! [Xd: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xd)),aa_int_int(times_times_int(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3188_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3189_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3190_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3191_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3192_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3193_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3194_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3195_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3196_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3197_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono
tff(fact_3198_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3199_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3200_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3201_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3202_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3203_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3204_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3205_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3206_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xd: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3207_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xd: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3208_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3209_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xd)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(Xd,Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3210_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3211_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3212_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_3213_power__Suc__le__self,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3214_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_3215_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_3216_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_3217_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3218_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3219_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3220_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3221_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3222_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3223_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3224_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3225_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3226_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3227_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_3228_self__le__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% self_le_power
tff(fact_3229_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_3230_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_3231_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_3232_dvd__power__iff,axiom,
    ! [Xd: nat,M: nat,Nb: nat] :
      ( ( Xd != 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,Xd),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xd),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3233_dvd__power__iff,axiom,
    ! [Xd: int,M: nat,Nb: nat] :
      ( ( Xd != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xd),M)),aa_nat_int(power_power_int(Xd),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3234_dvd__power__iff,axiom,
    ! [Xd: code_integer,M: nat,Nb: nat] :
      ( ( Xd != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xd),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3235_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_3236_power2__nat__le__eq__le,axiom,
    ! [M: 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,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),numeral_numeral_nat(bit0(one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_3237_power2__nat__le__imp__le,axiom,
    ! [M: 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,M),numeral_numeral_nat(bit0(one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_3238_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,M: 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)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),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,M),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_3239_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: 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,M),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,M),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3240_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( 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,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)) ) ) ).

% td_gal
tff(fact_3241_power__dvd__imp__le,axiom,
    ! [I: nat,M: 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),M)),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,M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_3242_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_3243_pos__mult__pos__ge,axiom,
    ! [Xd: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Nb),one_one_int)),aa_int_int(times_times_int(Nb),Xd)) ) ) ).

% pos_mult_pos_ge
tff(fact_3244_q__pos__lemma,axiom,
    ! [B5: int,Q6: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B5),Q6)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_3245_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B5),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B5),Q6)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
         => ( 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),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3246_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(plus_plus_int(X),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(plus_plus_int(X4),aa_int_int(times_times_int(K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3247_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B5),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B5),Q6)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3248_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3249_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3250_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(plus_plus_int(K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3251_int__mod__ge_H,axiom,
    ! [B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B),Nb)),aa_int_int(modulo_modulo_int(B),Nb)) ) ) ).

% int_mod_ge'
tff(fact_3252_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_3253_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,bit0(one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_3254_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(M),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3255_convex__bound__lt,axiom,
    ! [Xd: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xd),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V)
           => ( ( plus_p5714425477246183910nteger(U,V) = one_one_Code_integer )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(U),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3256_convex__bound__lt,axiom,
    ! [Xd: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3257_convex__bound__lt,axiom,
    ! [Xd: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),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)
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),Xd),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3258_convex__bound__lt,axiom,
    ! [Xd: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xd)),aa_int_int(times_times_int(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3259_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3260_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3261_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3262_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3263_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),numeral_numeral_nat(bit0(one)))) ).

% zero_le_power2
tff(fact_3264_zero__le__power2,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one)))) ).

% zero_le_power2
tff(fact_3265_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),numeral_numeral_nat(bit0(one)))) ).

% zero_le_power2
tff(fact_3266_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),numeral_numeral_nat(bit0(one)))) ).

% zero_le_power2
tff(fact_3267_power2__eq__imp__eq,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xd = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3268_power2__eq__imp__eq,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
         => ( Xd = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3269_power2__eq__imp__eq,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xd = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3270_power2__eq__imp__eq,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(bit0(one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xd)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xd = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3271_power2__eq__imp__eq,axiom,
    ! [Xd: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xd = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3272_power2__le__imp__le,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_le_imp_le
tff(fact_3273_power2__le__imp__le,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),Y) ) ) ).

% power2_le_imp_le
tff(fact_3274_power2__le__imp__le,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_le_imp_le
tff(fact_3275_power2__le__imp__le,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_le_imp_le
tff(fact_3276_power2__le__imp__le,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_le_imp_le
tff(fact_3277_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3278_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3279_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3280_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3281_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3282_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3283_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3284_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3285_num_Osize__gen_I1_J,axiom,
    size_num(one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3286_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3287_power__mono__odd,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3288_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3289_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3290_pos__mod__sign2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% pos_mod_sign2
tff(fact_3291_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q5: 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),Q5)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_3292_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,bit0(one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3293_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(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),M)),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,M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3294_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)) ).

% two_realpow_ge_one
tff(fact_3295_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
             => aa_int_o(P,J2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
             => aa_int_o(P,J2) ) ) ) ) ).

% split_zmod
tff(fact_3296_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3297_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3298_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3299_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(divide_divide_int(A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3300_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(divide_divide_int(Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J2: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_3301_mod__power__lem,axiom,
    ! [A: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),zero_zero_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).

% mod_power_lem
tff(fact_3302_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3303_power2__less__imp__less,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xd),Y) ) ) ).

% power2_less_imp_less
tff(fact_3304_power2__less__imp__less,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_less_imp_less
tff(fact_3305_power2__less__imp__less,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_less_imp_less
tff(fact_3306_power2__less__imp__less,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_less_imp_less
tff(fact_3307_power2__less__imp__less,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(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,Xd),Y) ) ) ).

% power2_less_imp_less
tff(fact_3308_sum__power2__ge__zero,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_3309_sum__power2__ge__zero,axiom,
    ! [Xd: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_3310_sum__power2__ge__zero,axiom,
    ! [Xd: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_3311_sum__power2__ge__zero,axiom,
    ! [Xd: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))) ).

% sum_power2_ge_zero
tff(fact_3312_sum__power2__le__zero__iff,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),zero_zero_real)
    <=> ( ( Xd = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3313_sum__power2__le__zero__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))),zero_z3403309356797280102nteger)
    <=> ( ( Xd = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3314_sum__power2__le__zero__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))))),zero_zero_rat)
    <=> ( ( Xd = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3315_sum__power2__le__zero__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),zero_zero_int)
    <=> ( ( Xd = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3316_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,numeral_numeral_nat(bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_3317_zero__le__even__power_H,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_3318_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,numeral_numeral_nat(bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_3319_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,numeral_numeral_nat(bit0(one))),Nb))) ).

% zero_le_even_power'
tff(fact_3320_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,numeral_numeral_nat(bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3321_zero__le__power__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3322_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,numeral_numeral_nat(bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3323_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,numeral_numeral_nat(bit0(one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3324_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3325_zero__le__odd__power,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ).

% zero_le_odd_power
tff(fact_3326_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3327_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3328_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3329_zero__le__even__power,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3330_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3331_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3332_two__pow__div__gt__le,axiom,
    ! [V: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_3333_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),numeral_numeral_nat(bit0(one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3334_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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,A2),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_3335_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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(plus_plus_int(aa_int_int(divide_divide_int(A2),Nb)),
                $ite(aa_int_int(modulo_modulo_int(B3),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(divide_divide_int(B3),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_3336_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J2),K)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) ) ) ).

% split_pos_lemma
tff(fact_3337_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I2: int,J2: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),zero_zero_int)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J2) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) ) ) ).

% split_neg_lemma
tff(fact_3338_sum__squares__bound,axiom,
    ! [Xd: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) ).

% sum_squares_bound
tff(fact_3339_sum__squares__bound,axiom,
    ! [Xd: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Xd)),Y)),plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))))) ).

% sum_squares_bound
tff(fact_3340_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,numeral_numeral_nat(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_3341_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3342_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,numeral_numeral_nat(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_3343_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,numeral_numeral_nat(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_3344_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_3345_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_3346_mod__double__modulus,axiom,
    ! [M: code_integer,Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
       => ( ( modulo364778990260209775nteger(Xd,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),M)) = modulo364778990260209775nteger(Xd,M) )
          | ( modulo364778990260209775nteger(Xd,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xd,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3347_mod__double__modulus,axiom,
    ! [M: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xd)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3348_mod__double__modulus,axiom,
    ! [M: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
       => ( ( aa_int_int(modulo_modulo_int(Xd),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(modulo_modulo_int(Xd),M) )
          | ( aa_int_int(modulo_modulo_int(Xd),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(Xd),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_3349_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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3350_power__le__zero__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),zero_z3403309356797280102nteger)
    <=> ( 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,numeral_numeral_nat(bit0(one))),Nb)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3351_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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3352_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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3353_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_3354_power__2__mult__step__le,axiom,
    ! [N3: nat,Nb: nat,K4: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),Nb)
     => ( 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(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N3)),K4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),K))
       => 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(power_power_nat,numeral_numeral_nat(bit0(one))),N3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),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,numeral_numeral_nat(bit0(one))),Nb)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_3355_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3356_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3357_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3358_eme1p,axiom,
    ! [Nb: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),Nb)),D) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(Nb),D)) ) ) ) ) ).

% eme1p
tff(fact_3359_emep1,axiom,
    ! [Nb: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Nb),one_one_int)),D) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(Nb),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_3360_count__buildup,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb))) ).

% count_buildup
tff(fact_3361_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_3362_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)) ).

% cnt_non_neg
tff(fact_3363_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A)) ) ).

% sb_dec_lem
tff(fact_3364_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_3365_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3366_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3367_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3368_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3369_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_3370_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(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_3371_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3372_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3373_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3374_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3375_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3376_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_3377_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_3378_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(A)) = A ).

% add.inverse_inverse
tff(fact_3379_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_3380_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_3381_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_3382_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_3383_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(B)) = B ).

% verit_minus_simplify(4)
tff(fact_3384_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_3385_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : uminus_uminus_rat(uminus_uminus_rat(B)) = B ).

% verit_minus_simplify(4)
tff(fact_3386_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_3387_semiring__norm_I90_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(90)
tff(fact_3388_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3389_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3390_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3391_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3392_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_3393_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3394_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_3395_diff__self,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% diff_self
tff(fact_3396_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_3397_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3398_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_3399_diff__0__right,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3400_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_3401_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_3402_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3403_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_3404_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_3405_diff__zero,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),zero_zero_int) = A ).

% diff_zero
tff(fact_3406_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3407_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3408_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_3409_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_3410_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3411_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_3412_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_3413_add_Oinverse__neutral,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_3414_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_3415_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_3416_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_3417_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_3418_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = uminus1351360451143612070nteger(A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3419_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_3420_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3421_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_3422_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_3423_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_3424_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_3425_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_3426_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_3427_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_3428_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = uminus1351360451143612070nteger(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_3429_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_3430_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_3431_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_3432_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_3433_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_3434_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_3435_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_3436_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_3437_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_3438_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel
tff(fact_3439_add__diff__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3440_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3441_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3442_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(minus_minus_rat(A,B),B) = A ).

% diff_add_cancel
tff(fact_3443_diff__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3444_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3445_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3446_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(plus_plus_rat(C,A),plus_plus_rat(C,B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_3447_add__diff__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(C),A)),aa_complex_complex(plus_plus_complex(C),B)) = aa_complex_complex(minus_minus_complex(A),B) ).

% add_diff_cancel_left
tff(fact_3448_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_3449_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3450_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C),A)),aa_int_int(plus_plus_int(C),B)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_left
tff(fact_3451_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),A) = B ).

% add_diff_cancel_left'
tff(fact_3452_add__diff__cancel__left_H,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3453_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3454_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3455_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3456_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_3457_add__diff__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(minus_minus_complex(A),B) ).

% add_diff_cancel_right
tff(fact_3458_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_3459_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3460_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(A),B) ).

% add_diff_cancel_right
tff(fact_3461_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).

% add_diff_cancel_right'
tff(fact_3462_add__diff__cancel__right_H,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3463_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3464_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3465_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3466_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_3467_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_3468_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_3469_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_3470_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3471_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3472_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3473_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3474_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3475_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(A),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_3476_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_3477_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B)) = B ).

% add_minus_cancel
tff(fact_3478_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_3479_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(uminus_uminus_rat(A),B)) = B ).

% add_minus_cancel
tff(fact_3480_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(plus_plus_int(A),B)) = B ).

% minus_add_cancel
tff(fact_3481_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_3482_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_3483_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_3484_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),plus_plus_rat(A,B)) = B ).

% minus_add_cancel
tff(fact_3485_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_3486_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_3487_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_add_distrib
tff(fact_3488_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_3489_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) ).

% minus_add_distrib
tff(fact_3490_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_right
tff(fact_3491_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_3492_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ).

% mult_minus_right
tff(fact_3493_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_3494_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_3495_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(times_times_int(A),B) ).

% minus_mult_minus
tff(fact_3496_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_3497_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),uminus1351360451143612070nteger(B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B) ).

% minus_mult_minus
tff(fact_3498_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_3499_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_3500_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A),B)) ).

% mult_minus_left
tff(fact_3501_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_3502_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),B) = uminus1351360451143612070nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) ).

% mult_minus_left
tff(fact_3503_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_3504_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_3505_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(minus_minus_int(B),A) ).

% minus_diff_eq
tff(fact_3506_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_3507_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_3508_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_3509_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_3510_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : bit0(M) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_3511_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(bit1,M) != bit0(Nb) ).

% semiring_norm(89)
tff(fact_3512_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_3513_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_3514_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(A),B) ).

% div_minus_minus
tff(fact_3515_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_3516_dvd__minus__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Y) ) ).

% dvd_minus_iff
tff(fact_3517_dvd__minus__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Y) ) ).

% dvd_minus_iff
tff(fact_3518_dvd__minus__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),uminus1351360451143612070nteger(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),Y) ) ).

% dvd_minus_iff
tff(fact_3519_dvd__minus__iff,axiom,
    ! [Xd: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xd),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(dvd_dvd_complex(Xd),Y) ) ).

% dvd_minus_iff
tff(fact_3520_dvd__minus__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xd),uminus_uminus_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xd),Y) ) ).

% dvd_minus_iff
tff(fact_3521_minus__dvd__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xd)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Y) ) ).

% minus_dvd_iff
tff(fact_3522_minus__dvd__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xd)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Y) ) ).

% minus_dvd_iff
tff(fact_3523_minus__dvd__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,uminus1351360451143612070nteger(Xd)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xd),Y) ) ).

% minus_dvd_iff
tff(fact_3524_minus__dvd__iff,axiom,
    ! [Xd: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),Y)
    <=> aa_complex_o(dvd_dvd_complex(Xd),Y) ) ).

% minus_dvd_iff
tff(fact_3525_minus__dvd__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,uminus_uminus_rat(Xd)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xd),Y) ) ).

% minus_dvd_iff
tff(fact_3526_Suc__diff__diff,axiom,
    ! [M: 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,M)),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,M),Nb)),K) ).

% Suc_diff_diff
tff(fact_3527_diff__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% diff_Suc_Suc
tff(fact_3528_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_3529_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_3530_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3531_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_3532_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_3533_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_3534_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_3535_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_3536_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(73)
tff(fact_3537_semiring__norm_I80_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(80)
tff(fact_3538_count__buildup_H,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)))) ).

% count_buildup'
tff(fact_3539_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3540_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3541_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3542_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3543_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3544_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_3545_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3546_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3547_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3548_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3549_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3550_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_3551_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_3552_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_3553_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_3554_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_3555_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_3556_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_3557_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_3558_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_3559_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_3560_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_3561_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_3562_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_3563_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_3564_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_3565_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_3566_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_3567_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3568_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3569_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_3570_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_3571_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3572_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_3573_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_3574_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_3575_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_3576_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_3577_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_3578_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_3579_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_3580_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_3581_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_3582_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_3583_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_3584_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_3585_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_3586_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_3587_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_3588_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3589_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( plus_plus_rat(minus_minus_rat(A,B),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3590_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3591_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3592_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3593_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3594_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3595_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3596_right__diff__distrib__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3597_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3598_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3599_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3600_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3601_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),numera6620942414471956472nteger(V)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),numera6620942414471956472nteger(V))) ).

% left_diff_distrib_numeral
tff(fact_3602_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_3603_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_3604_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_3605_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_3606_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_3607_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_3608_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_3609_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_3610_add_Oright__inverse,axiom,
    ! [A: rat] : plus_plus_rat(A,uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_3611_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_3612_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_3613_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_3614_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_3615_ab__left__minus,axiom,
    ! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_3616_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(minus_minus_int(zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_3617_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_3618_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = uminus1351360451143612070nteger(B) ).

% verit_minus_simplify(3)
tff(fact_3619_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_3620_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).

% verit_minus_simplify(3)
tff(fact_3621_diff__0,axiom,
    ! [A: int] : aa_int_int(minus_minus_int(zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_3622_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_3623_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = uminus1351360451143612070nteger(A) ).

% diff_0
tff(fact_3624_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_3625_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_3626_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3627_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3628_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = uminus1351360451143612070nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3629_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3630_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Nb))) = uminus_uminus_rat(plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3631_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_3632_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_3633_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(Z) ).

% mult_minus1_right
tff(fact_3634_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_3635_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_3636_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_3637_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_3638_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Z) = uminus1351360451143612070nteger(Z) ).

% mult_minus1
tff(fact_3639_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_3640_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_3641_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(minus_minus_int(B),A) ).

% uminus_add_conv_diff
tff(fact_3642_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_3643_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_3644_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_3645_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),B) = minus_minus_rat(B,A) ).

% uminus_add_conv_diff
tff(fact_3646_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(plus_plus_int(A),B) ).

% diff_minus_eq_add
tff(fact_3647_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_3648_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,uminus1351360451143612070nteger(B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_3649_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_3650_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,uminus_uminus_rat(B)) = plus_plus_rat(A,B) ).

% diff_minus_eq_add
tff(fact_3651_divide__minus1,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xd) ).

% divide_minus1
tff(fact_3652_divide__minus1,axiom,
    ! [Xd: complex] : aa_complex_complex(divide1717551699836669952omplex(Xd),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xd) ).

% divide_minus1
tff(fact_3653_divide__minus1,axiom,
    ! [Xd: rat] : divide_divide_rat(Xd,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Xd) ).

% divide_minus1
tff(fact_3654_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_3655_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(A) ).

% div_minus1_right
tff(fact_3656_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3657_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3658_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3659_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3660_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3661_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_3662_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_3663_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_3664_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_3665_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_3666_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_3667_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_3668_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_3669_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_3670_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_3671_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,numeral_numeral_nat(Nb)) = numeral_numeral_rat(Nb) ).

% of_nat_numeral
tff(fact_3672_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,numeral_numeral_nat(Nb)) = numeral_numeral_real(Nb) ).

% of_nat_numeral
tff(fact_3673_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,numeral_numeral_nat(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_3674_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,numeral_numeral_nat(Nb)) = numeral_numeral_nat(Nb) ).

% of_nat_numeral
tff(fact_3675_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,numeral_numeral_nat(Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_3676_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3677_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3678_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3679_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3680_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3681_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3682_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3683_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3684_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_3685_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_3686_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_3687_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_3688_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_3689_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_add
tff(fact_3690_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_3691_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% minus_mod_self1
tff(fact_3692_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_3693_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_3694_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_3695_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_3696_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_3697_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_3698_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_3699_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_3700_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_3701_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_3702_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3703_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3704_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3705_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3706_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3707_zero__less__diff,axiom,
    ! [Nb: nat,M: 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),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% zero_less_diff
tff(fact_3708_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_3709_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3710_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% diff_is_0_eq
tff(fact_3711_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_3712_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_3713_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_3714_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_3715_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_3716_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_3717_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_3718_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_3719_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xd) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3720_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xd) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3721_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xd) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3722_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xd) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3723_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xd) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3724_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xd) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xd ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3725_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xd) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xd ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3726_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xd) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xd ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3727_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xd) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xd ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3728_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xd) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xd ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3729_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3730_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3731_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3732_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3733_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3734_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_3735_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_3736_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_3737_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_3738_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_real(zero_n3304061248610475627l_real,~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_o_real(zero_n3304061248610475627l_real,(P))) ).

% of_bool_not_iff
tff(fact_3739_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(minus_minus_int(one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_3740_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_3741_semiring__norm_I9_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),bit0(Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(9)
tff(fact_3742_semiring__norm_I7_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% semiring_norm(7)
tff(fact_3743_semiring__norm_I15_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),bit0(Nb)) = bit0(aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),Nb)) ).

% semiring_norm(15)
tff(fact_3744_semiring__norm_I14_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,bit0(M)),aa_num_num(bit1,Nb)) = bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_3745_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(bit0(M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(72)
tff(fact_3746_semiring__norm_I81_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(aa_num_num(bit1,M)),bit0(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(81)
tff(fact_3747_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_3748_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(ord_less_num(one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_3749_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% of_nat_of_bool
tff(fact_3750_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_complex(semiri8010041392384452111omplex,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n1201886186963655149omplex((P)) ).

% of_nat_of_bool
tff(fact_3751_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_3752_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_3753_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_3754_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_3755_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).

% dbl_simps(1)
tff(fact_3756_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K))) = uminus1351360451143612070nteger(neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_3757_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_3758_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_numeral_dbl_rat(numeral_numeral_rat(K))) ).

% dbl_simps(1)
tff(fact_3759_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_3760_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_3761_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_3762_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_3763_add__neg__numeral__special_I8_J,axiom,
    plus_plus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_3764_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(plus_plus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_3765_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_3766_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_3767_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_3768_add__neg__numeral__special_I7_J,axiom,
    plus_plus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_3769_diff__numeral__special_I12_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_3770_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_3771_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_3772_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_3773_diff__numeral__special_I12_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_3774_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_3775_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3776_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3777_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_3778_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(Nb)) = uminus_uminus_rat(one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3779_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_3780_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3781_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( uminus1351360451143612070nteger(one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3782_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_3783_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3784_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3785_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3786_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3787_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3788_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3789_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_3790_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3791_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_3792_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3793_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_3794_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_3795_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_3796_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_3797_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_3798_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = plus_plus_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3799_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_3800_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3801_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_3802_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_3803_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_3804_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_3805_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_3806_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_3807_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_3808_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3809_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_3810_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3811_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3812_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_3813_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3814_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(W)),Y)) = plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_3815_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3816_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3817_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3818_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3819_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3820_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3821_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3822_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3823_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3824_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(Nb)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3825_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bit0(Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_3826_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_3827_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bit0(M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_3828_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_3829_semiring__norm_I10_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = bit0(aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)),one)) ).

% semiring_norm(10)
tff(fact_3830_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3831_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3832_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3833_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3834_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3835_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3836_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3837_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3838_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3839_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Nb)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3840_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3841_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3842_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3843_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3844_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Nb))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3845_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3846_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3847_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3848_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3849_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3850_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3851_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3852_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3853_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3854_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3855_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_3856_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_3857_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_3858_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_3859_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_3860_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(minus_minus_int(Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_3861_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_3862_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_3863_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_3864_neg__numeral__le__iff,axiom,
    ! [M: 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,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(ord_less_eq_num(Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_3865_neg__numeral__less__iff,axiom,
    ! [M: 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,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(ord_less_num(Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_3866_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(ord_less_num(Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_3867_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(ord_less_num(Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_3868_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(Nb)))
    <=> aa_num_o(ord_less_num(Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_3869_semiring__norm_I16_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Nb)),bit0(aa_num_num(aa_num_fun_num_num(times_times_num,M),Nb)))) ).

% semiring_norm(16)
tff(fact_3870_semiring__norm_I79_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_num(bit0(M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(ord_less_eq_num(M),Nb) ) ).

% semiring_norm(79)
tff(fact_3871_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(ord_less_eq_num(aa_num_num(bit1,M)),bit0(Nb))
    <=> aa_num_o(ord_less_num(M),Nb) ) ).

% semiring_norm(74)
tff(fact_3872_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3873_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3874_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3875_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3876_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3877_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3878_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3879_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3880_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3881_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3882_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3883_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3884_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3885_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3886_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3887_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3888_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3889_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3890_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3891_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3892_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3893_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3894_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) ).

% power2_minus
tff(fact_3895_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one))) ).

% power2_minus
tff(fact_3896_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) ).

% power2_minus
tff(fact_3897_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),numeral_numeral_nat(bit0(one))) = aa_nat_complex(power_power_complex(A),numeral_numeral_nat(bit0(one))) ).

% power2_minus
tff(fact_3898_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),numeral_numeral_nat(bit0(one))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one))) ).

% power2_minus
tff(fact_3899_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_3900_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_3901_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_3902_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_3903_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_3904_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3905_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3906_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3907_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3908_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3909_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xd: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xd)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3910_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3911_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3912_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3913_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3914_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3915_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xd)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3916_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3917_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xd)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3918_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3919_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xd)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3920_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3921_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3922_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3923_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3924_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3925_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3926_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3927_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3928_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xd)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3929_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3930_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xd: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xd)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3931_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3932_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3933_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3934_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3935_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xd: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xd) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3936_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),numeral_numeral_nat(W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_3937_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_3938_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3939_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_3940_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3941_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3942_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3943_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(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3944_add__neg__numeral__special_I9_J,axiom,
    plus_plus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3945_diff__numeral__special_I10_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3946_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3947_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3948_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(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3949_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3950_diff__numeral__special_I11_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3951_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3952_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = numera6620942414471956472nteger(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3953_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3954_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3955_minus__1__div__2__eq,axiom,
    aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_3956_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_3957_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(minus_minus_int(A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(A),B)) ) ).

% even_diff
tff(fact_3958_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_3959_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_3960_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_3961_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_3962_of__nat__zero__less__power__iff,axiom,
    ! [Xd: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3963_of__nat__zero__less__power__iff,axiom,
    ! [Xd: 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,Xd)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3964_of__nat__zero__less__power__iff,axiom,
    ! [Xd: 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,Xd)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3965_of__nat__zero__less__power__iff,axiom,
    ! [Xd: 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,Xd)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3966_of__nat__zero__less__power__iff,axiom,
    ! [Xd: 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,Xd)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3967_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_3968_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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_3969_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_3970_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,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_3971_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_3972_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3973_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3974_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_3975_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3976_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_3977_power__minus__odd,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3978_power__minus__odd,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3979_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_3980_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3981_power__minus__odd,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_3982_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),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,numeral_numeral_nat(aa_num_num(bit1,one))),M)),numeral_numeral_nat(V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_3983_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_3984_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),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,numeral_numeral_nat(aa_num_num(bit1,one))),M)),numeral_numeral_nat(V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_3985_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_3986_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_3987_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_3988_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_3989_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_3990_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_3991_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3992_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3993_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3994_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3995_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3996_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,bit0(one))) ).

% dbl_simps(4)
tff(fact_3997_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one))) ).

% dbl_simps(4)
tff(fact_3998_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(bit0(one))) ).

% dbl_simps(4)
tff(fact_3999_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(bit0(one))) ).

% dbl_simps(4)
tff(fact_4000_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% dbl_simps(4)
tff(fact_4001_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,numeral_numeral_nat(bit0(one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_4002_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,numeral_numeral_nat(bit0(one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_4003_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4004_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,numeral_numeral_nat(bit0(one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_4005_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_4006_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4007_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4008_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4009_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4010_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4011_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4012_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4013_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4014_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4015_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4016_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4017_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xd)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4018_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4019_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xd)),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,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4020_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4021_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4022_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4023_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4024_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: 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,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4025_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4026_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4027_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4028_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xd)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4029_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4030_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xd: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xd)),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,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4031_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4032_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4033_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4034_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4035_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xd: 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,Xd))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(I)),Nb)),Xd) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4036_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ) ).

% even_of_nat
tff(fact_4037_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ) ).

% even_of_nat
tff(fact_4038_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_4039_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% signed_take_bit_0
tff(fact_4040_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% signed_take_bit_0
tff(fact_4041_even__diff__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ) ) ).

% even_diff_nat
tff(fact_4042_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,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4043_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4044_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4045_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,bit0(one))),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4046_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4047_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_4048_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_4049_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(minus_minus_int(zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_4050_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_4051_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_4052_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_4053_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ) ) ).

% of_nat_diff
tff(fact_4054_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_4055_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_4056_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% eq_imp_le
tff(fact_4057_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_4058_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( 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,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( M = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_4059_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( 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,M),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,M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_4060_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( 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,M),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,M),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4061_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_4062_diff__le__self,axiom,
    ! [M: 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,M),Nb)),M) ).

% diff_le_self
tff(fact_4063_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4064_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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),M)) ) ).

% diff_le_mono2
tff(fact_4065_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% nat_le_linear
tff(fact_4066_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),S3)
     => ( ? [Z3: real] :
          ! [X: real] :
            ( aa_set_real_o(member_real(X),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z3) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( aa_set_real_o(member_real(X4),S3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y2) )
            & ! [Z3: real] :
                ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),S3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z3) ) ) ) ) ).

% complete_real
tff(fact_4067_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B) )
       => ? [X: nat] :
            ( aa_nat_o(P,X)
            & ! [Y3: nat] :
                ( aa_nat_o(P,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_4068_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% signed_take_bit_diff
tff(fact_4069_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_4070_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4071_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4072_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4073_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4074_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_4075_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_4076_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_4077_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,uminus1351360451143612070nteger(B)) ).

% diff_conv_add_uminus
tff(fact_4078_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_4079_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_4080_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = aa_int_int(plus_plus_int(K),B) )
     => ( aa_int_int(minus_minus_int(A),B3) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4081_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( 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),B)) ) ) ).

% group_cancel.sub2
tff(fact_4082_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B3) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_4083_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( 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),B)) ) ) ).

% group_cancel.sub2
tff(fact_4084_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = plus_plus_rat(K,B) )
     => ( minus_minus_rat(A,B3) = plus_plus_rat(uminus_uminus_rat(K),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_4085_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: complex,C: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),C)),B) = aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4086_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4087_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4088_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_4089_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_4090_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_4091_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(B),A) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),B) ).

% minus_diff_commute
tff(fact_4092_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_4093_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : minus_minus_rat(uminus_uminus_rat(B),A) = minus_minus_rat(uminus_uminus_rat(A),B) ).

% minus_diff_commute
tff(fact_4094_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_4095_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_4096_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( uminus1351360451143612070nteger(B) = A ) ) ).

% minus_equation_iff
tff(fact_4097_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_4098_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_4099_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_4100_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_4101_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% equation_minus_iff
tff(fact_4102_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_4103_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_4104_diff__eq__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(minus_minus_complex(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4105_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4106_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4107_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4108_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4109_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( uminus1351360451143612070nteger(A) = uminus1351360451143612070nteger(B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4110_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( uminus_uminus_rat(A) = uminus_uminus_rat(B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4111_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A),B)) ).

% minus_diff_minus
tff(fact_4112_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_4113_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(A),uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_4114_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_4115_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(minus_minus_rat(A,B)) ).

% minus_diff_minus
tff(fact_4116_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_4117_power__minus__Bit1,axiom,
    ! [Xd: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xd)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4118_power__minus__Bit1,axiom,
    ! [Xd: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xd)),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,Xd),numeral_numeral_nat(aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4119_power__minus__Bit1,axiom,
    ! [Xd: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(Xd)),numeral_numeral_nat(aa_num_num(bit1,K))) = uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4120_power__minus__Bit1,axiom,
    ! [Xd: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),numeral_numeral_nat(aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4121_power__minus__Bit1,axiom,
    ! [Xd: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xd)),numeral_numeral_nat(aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4122_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B),aa_int_int(modulo_modulo_int(A),B))) ).

% zmod_zminus1_eq_if
tff(fact_4123_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_4124_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_4125_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_4126_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_4127_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_4128_mult__of__nat__commute,axiom,
    ! [Xd: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xd)) ).

% mult_of_nat_commute
tff(fact_4129_mult__of__nat__commute,axiom,
    ! [Xd: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xd)) ).

% mult_of_nat_commute
tff(fact_4130_mult__of__nat__commute,axiom,
    ! [Xd: nat,Y: int] : aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Xd)),Y) = aa_int_int(times_times_int(Y),aa_nat_int(semiri1314217659103216013at_int,Xd)) ).

% mult_of_nat_commute
tff(fact_4131_mult__of__nat__commute,axiom,
    ! [Xd: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xd)) ).

% mult_of_nat_commute
tff(fact_4132_mult__of__nat__commute,axiom,
    ! [Xd: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xd)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,Xd)) ).

% mult_of_nat_commute
tff(fact_4133_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4134_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_4135_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4136_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(minus_minus_int(A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_4137_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4138_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4139_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4140_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_4141_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_right_mono
tff(fact_4142_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_right_mono
tff(fact_4143_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_4144_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_left_mono
tff(fact_4145_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_left_mono
tff(fact_4146_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_4147_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_mono
tff(fact_4148_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D)) ) ) ).

% diff_mono
tff(fact_4149_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4150_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_strict_mono
tff(fact_4151_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4152_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4153_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4154_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = aa_int_int(minus_minus_int(C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4155_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4156_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_strict_left_mono
tff(fact_4157_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(C),A)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4158_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4159_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_strict_right_mono
tff(fact_4160_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4161_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_4162_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)) ) ).

% le_imp_neg_le
tff(fact_4163_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% le_imp_neg_le
tff(fact_4164_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_4165_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_4166_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B)),A) ) ).

% minus_le_iff
tff(fact_4167_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),A) ) ).

% minus_le_iff
tff(fact_4168_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_4169_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_4170_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),uminus1351360451143612070nteger(B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),uminus1351360451143612070nteger(A)) ) ).

% le_minus_iff
tff(fact_4171_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),uminus_uminus_rat(A)) ) ).

% le_minus_iff
tff(fact_4172_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_4173_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,D)) = plus_plus_rat(minus_minus_rat(A,B),minus_minus_rat(C,D)) ).

% add_diff_add
tff(fact_4174_add__diff__add,axiom,
    ! [A: complex,C: complex,B: complex,D: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),aa_complex_complex(plus_plus_complex(B),D)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),aa_complex_complex(minus_minus_complex(C),D)) ).

% add_diff_add
tff(fact_4175_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_4176_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),aa_int_int(plus_plus_int(B),D)) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),aa_int_int(minus_minus_int(C),D)) ).

% add_diff_add
tff(fact_4177_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( minus_minus_rat(A2,B) = plus_plus_rat(K,minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_4178_group__cancel_Osub1,axiom,
    ! [A2: complex,K: complex,A: complex,B: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(minus_minus_complex(A2),B) = aa_complex_complex(plus_plus_complex(K),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4179_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4180_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(minus_minus_int(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4181_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = plus_plus_rat(C,B) ) ) ).

% diff_eq_eq
tff(fact_4182_diff__eq__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( aa_complex_complex(minus_minus_complex(A),B) = C )
    <=> ( A = aa_complex_complex(plus_plus_complex(C),B) ) ) ).

% diff_eq_eq
tff(fact_4183_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_4184_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(minus_minus_int(A),B) = C )
    <=> ( A = aa_int_int(plus_plus_int(C),B) ) ) ).

% diff_eq_eq
tff(fact_4185_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( plus_plus_rat(A,B) = C ) ) ).

% eq_diff_eq
tff(fact_4186_eq__diff__eq,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( A = aa_complex_complex(minus_minus_complex(C),B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4187_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4188_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(minus_minus_int(C),B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4189_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,B),C) ).

% add_diff_eq
tff(fact_4190_add__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),C) ).

% add_diff_eq
tff(fact_4191_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_4192_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),C) ).

% add_diff_eq
tff(fact_4193_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,C),B) ).

% diff_diff_eq2
tff(fact_4194_diff__diff__eq2,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),B) ).

% diff_diff_eq2
tff(fact_4195_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_4196_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_diff_eq2
tff(fact_4197_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : plus_plus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(plus_plus_rat(A,C),B) ).

% diff_add_eq
tff(fact_4198_diff__add__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),B) ).

% diff_add_eq
tff(fact_4199_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_4200_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),C)),B) ).

% diff_add_eq
tff(fact_4201_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,plus_plus_rat(B,C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4202_diff__add__eq__diff__diff__swap,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4203_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4204_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4205_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( plus_plus_rat(C,B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_4206_add__implies__diff,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(C),B) = A )
     => ( C = aa_complex_complex(minus_minus_complex(A),B) ) ) ).

% add_implies_diff
tff(fact_4207_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_4208_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4209_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(plus_plus_int(C),B) = A )
     => ( C = aa_int_int(minus_minus_int(A),B) ) ) ).

% add_implies_diff
tff(fact_4210_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,plus_plus_rat(B,C)) ).

% diff_diff_eq
tff(fact_4211_diff__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(A),aa_complex_complex(plus_plus_complex(B),C)) ).

% diff_diff_eq
tff(fact_4212_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_4213_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_4214_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(A),aa_int_int(plus_plus_int(B),C)) ).

% diff_diff_eq
tff(fact_4215_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X: real,K2: real] :
          ( aa_real_o(P,X)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D3))) )
     => ( ! [X: real,K2: real] :
            ( aa_real_o(Q,X)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D3))) )
       => ! [X4: real,K5: 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,K5),D3)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4216_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X: int,K2: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
     => ( ! [X: int,K2: int] :
            ( aa_int_o(Q,X)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
       => ! [X4: int,K5: int] :
            ( ( aa_int_o(P,X4)
              & aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3)))
              & aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4217_inf__period_I1_J,axiom,
    ! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
      ( ! [X: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(P,X)
        <=> aa_Code_integer_o(P,minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K2),D3))) )
     => ( ! [X: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Q,X)
          <=> aa_Code_integer_o(Q,minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K2),D3))) )
       => ! [X4: code_integer,K5: code_integer] :
            ( ( aa_Code_integer_o(P,X4)
              & aa_Code_integer_o(Q,X4) )
          <=> ( aa_Code_integer_o(P,minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3)))
              & aa_Code_integer_o(Q,minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4218_inf__period_I1_J,axiom,
    ! [P: fun_complex_o,D3: complex,Q: fun_complex_o] :
      ( ! [X: complex,K2: complex] :
          ( aa_complex_o(P,X)
        <=> aa_complex_o(P,aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K2),D3))) )
     => ( ! [X: complex,K2: complex] :
            ( aa_complex_o(Q,X)
          <=> aa_complex_o(Q,aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K2),D3))) )
       => ! [X4: complex,K5: complex] :
            ( ( aa_complex_o(P,X4)
              & aa_complex_o(Q,X4) )
          <=> ( aa_complex_o(P,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3)))
              & aa_complex_o(Q,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4219_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X: real,K2: real] :
          ( aa_real_o(P,X)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D3))) )
     => ( ! [X: real,K2: real] :
            ( aa_real_o(Q,X)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D3))) )
       => ! [X4: real,K5: 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,K5),D3)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4220_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X: int,K2: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
     => ( ! [X: int,K2: int] :
            ( aa_int_o(Q,X)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
       => ! [X4: int,K5: int] :
            ( ( aa_int_o(P,X4)
              | aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3)))
              | aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4221_inf__period_I2_J,axiom,
    ! [P: fun_Code_integer_o,D3: code_integer,Q: fun_Code_integer_o] :
      ( ! [X: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(P,X)
        <=> aa_Code_integer_o(P,minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K2),D3))) )
     => ( ! [X: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Q,X)
          <=> aa_Code_integer_o(Q,minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K2),D3))) )
       => ! [X4: code_integer,K5: code_integer] :
            ( ( aa_Code_integer_o(P,X4)
              | aa_Code_integer_o(Q,X4) )
          <=> ( aa_Code_integer_o(P,minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3)))
              | aa_Code_integer_o(Q,minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4222_inf__period_I2_J,axiom,
    ! [P: fun_complex_o,D3: complex,Q: fun_complex_o] :
      ( ! [X: complex,K2: complex] :
          ( aa_complex_o(P,X)
        <=> aa_complex_o(P,aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K2),D3))) )
     => ( ! [X: complex,K2: complex] :
            ( aa_complex_o(Q,X)
          <=> aa_complex_o(Q,aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(K2),D3))) )
       => ! [X4: complex,K5: complex] :
            ( ( aa_complex_o(P,X4)
              | aa_complex_o(Q,X4) )
          <=> ( aa_complex_o(P,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3)))
              | aa_complex_o(Q,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4223_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_4224_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_4225_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib'
tff(fact_4226_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)) ).

% right_diff_distrib'
tff(fact_4227_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib'
tff(fact_4228_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_4229_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_4230_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),C)),A) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(B),A)),aa_int_int(times_times_int(C),A)) ).

% left_diff_distrib'
tff(fact_4231_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,C)),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),A),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),A)) ).

% left_diff_distrib'
tff(fact_4232_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),C)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(B),A)),aa_complex_complex(times_times_complex(C),A)) ).

% left_diff_distrib'
tff(fact_4233_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_4234_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(A),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),B)),aa_int_int(times_times_int(A),C)) ).

% right_diff_distrib
tff(fact_4235_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C)) ).

% right_diff_distrib
tff(fact_4236_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib
tff(fact_4237_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_4238_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A),C)),aa_int_int(times_times_int(B),C)) ).

% left_diff_distrib
tff(fact_4239_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),C),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),C)) ).

% left_diff_distrib
tff(fact_4240_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% left_diff_distrib
tff(fact_4241_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4242_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4243_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),uminus1351360451143612070nteger(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4244_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_4245_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_4246_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_4247_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),uminus1351360451143612070nteger(B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),uminus1351360451143612070nteger(A)) ) ).

% less_minus_iff
tff(fact_4248_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),uminus_uminus_rat(A)) ) ).

% less_minus_iff
tff(fact_4249_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_4250_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_4251_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B)),A) ) ).

% minus_less_iff
tff(fact_4252_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),A) ) ).

% minus_less_iff
tff(fact_4253_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_4254_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_4255_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numera6620942414471956472nteger(M) != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_4256_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_4257_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_4258_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_4259_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(Nb) ).

% neg_numeral_neq_numeral
tff(fact_4260_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_4261_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_4262_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(Nb) ).

% neg_numeral_neq_numeral
tff(fact_4263_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_4264_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_4265_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% is_num_normalize(8)
tff(fact_4266_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_4267_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).

% is_num_normalize(8)
tff(fact_4268_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(plus_plus_int(K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4269_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_4270_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( uminus1351360451143612070nteger(A2) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),uminus1351360451143612070nteger(A)) ) ) ).

% group_cancel.neg1
tff(fact_4271_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_4272_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = plus_plus_rat(K,A) )
     => ( uminus_uminus_rat(A2) = plus_plus_rat(uminus_uminus_rat(K),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_4273_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_4274_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_4275_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B),uminus1351360451143612070nteger(A)) ).

% add.inverse_distrib_swap
tff(fact_4276_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_4277_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).

% add.inverse_distrib_swap
tff(fact_4278_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_4279_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_4280_one__neq__neg__one,axiom,
    one_one_Code_integer != uminus1351360451143612070nteger(one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_4281_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_4282_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_4283_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(times_times_int(A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_4284_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_4285_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(A)),B) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),uminus1351360451143612070nteger(B)) ).

% minus_mult_commute
tff(fact_4286_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_4287_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_4288_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(times_times_int(A),A) = aa_int_int(times_times_int(B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_4289_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_4290_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),B) )
    <=> ( ( A = B )
        | ( A = uminus1351360451143612070nteger(B) ) ) ) ).

% square_eq_iff
tff(fact_4291_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_4292_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_4293_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_4294_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_4295_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% diff_divide_distrib
tff(fact_4296_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_4297_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) ).

% div_minus_right
tff(fact_4298_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_4299_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_4300_minus__divide__left,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(uminus_uminus_rat(A),B) ).

% minus_divide_left
tff(fact_4301_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_4302_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_4303_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ).

% minus_divide_divide
tff(fact_4304_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_4305_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_4306_minus__divide__right,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ).

% minus_divide_right
tff(fact_4307_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : bit0(X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_4308_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_4309_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(minus_minus_int(B),C)) ) ).

% dvd_diff_commute
tff(fact_4310_dvd__diff,axiom,
    ! [Xd: complex,Y: complex,Z: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xd),Y)
     => ( aa_complex_o(dvd_dvd_complex(Xd),Z)
       => aa_complex_o(dvd_dvd_complex(Xd),aa_complex_complex(minus_minus_complex(Y),Z)) ) ) ).

% dvd_diff
tff(fact_4311_dvd__diff,axiom,
    ! [Xd: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_4312_dvd__diff,axiom,
    ! [Xd: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),aa_int_int(minus_minus_int(Y),Z)) ) ) ).

% dvd_diff
tff(fact_4313_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_4314_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% Suc_diff_le
tff(fact_4315_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_right_eq
tff(fact_4316_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_4317_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_left_eq
tff(fact_4318_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_4319_mod__diff__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B5: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A3),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B5),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A3),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4320_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B5),C) ) ) ) ).

% mod_diff_cong
tff(fact_4321_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A),B)),C) ).

% mod_diff_eq
tff(fact_4322_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_4323_diffs0__imp__equal,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M) = zero_zero_nat )
       => ( M = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_4324_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_4325_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_4326_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B)) ).

% mod_minus_right
tff(fact_4327_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: int,B: int,A3: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A3),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A3)),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_4328_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
     => ( modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A3),B) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_4329_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_4330_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A),B) ).

% mod_minus_eq
tff(fact_4331_diff__diff__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb) ) ) ).

% diff_diff_less
tff(fact_4332_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_4333_diff__less__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),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),M)) ) ) ).

% diff_less_mono2
tff(fact_4334_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_4335_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( 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,M),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,M),Nb) ) ) ) ).

% less_diff_iff
tff(fact_4336_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(minus_minus_int(K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_4337_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% Nat.diff_cancel
tff(fact_4338_diff__cancel2,axiom,
    ! [M: 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,M),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,M),Nb) ).

% diff_cancel2
tff(fact_4339_diff__add__inverse,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),Nb) = M ).

% diff_add_inverse
tff(fact_4340_diff__add__inverse2,axiom,
    ! [M: 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,M),Nb)),Nb) = M ).

% diff_add_inverse2
tff(fact_4341_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_4342_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_4343_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_4344_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_4345_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_4346_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_4347_diff__mult__distrib2,axiom,
    ! [K: nat,M: 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,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_4348_diff__mult__distrib,axiom,
    ! [M: 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,M),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_4349_Word_Omod__minus__cong,axiom,
    ! [B: int,B5: int,Xd: int,X6: int,Y: int,Y4: int,Z4: int] :
      ( ( B = B5 )
     => ( ( aa_int_int(modulo_modulo_int(Xd),B5) = aa_int_int(modulo_modulo_int(X6),B5) )
       => ( ( aa_int_int(modulo_modulo_int(Y),B5) = aa_int_int(modulo_modulo_int(Y4),B5) )
         => ( ( aa_int_int(minus_minus_int(X6),Y4) = Z4 )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(Xd),Y)),B) = aa_int_int(modulo_modulo_int(Z4),B5) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_4350_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( 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,M),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_4351_dvd__diffD,axiom,
    ! [K: nat,M: 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,M),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),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_4352_dvd__diffD1,axiom,
    ! [K: nat,M: 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,M),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_4353_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_4354_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_4355_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),T)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,D)),T) ) ).

% uminus_dvd_conv(1)
tff(fact_4356_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),T)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(uminus_uminus_int,T)) ) ).

% uminus_dvd_conv(2)
tff(fact_4357_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4358_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xd),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4359_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(Xd,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(Xd),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4360_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4361_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4362_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4363_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4364_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4365_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4366_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_4367_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_4368_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(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_4369_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_4370_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(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(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_4371_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_4372_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_4373_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_4374_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_4375_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_4376_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_4377_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_4378_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_4379_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_4380_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_4381_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_4382_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4383_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4384_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4385_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4386_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_4387_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_4388_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_4389_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_4390_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_4391_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_4392_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_4393_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_4394_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_4395_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% div_mult2_eq'
tff(fact_4396_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_int_int(divide_divide_int(A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_4397_div__mult2__eq_H,axiom,
    ! [A: nat,M: 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,M)),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,M))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_4398_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_4399_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(minus_minus_int(B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_4400_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,A),aa_nat_rat(semiri681578069525770553at_rat,B)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4401_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A)),aa_nat_real(semiri5074537144036343181t_real,B)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4402_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4403_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A)),aa_nat_nat(semiri1316708129612266289at_nat,B)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4404_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] : aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,A)),aa_nat_complex(semiri8010041392384452111omplex,B)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% Abs_fnat_hom_add
tff(fact_4405_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_4406_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4407_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_4408_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_4409_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4410_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_4411_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_4412_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(Nb,K)) ) ).

% add_le_imp_le_diff
tff(fact_4413_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_4414_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(plus_plus_int(I),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(minus_minus_int(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_4415_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_4416_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,plus_plus_rat(I,K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),plus_plus_rat(J,K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),plus_plus_rat(J,K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Nb,K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4417_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_4418_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(plus_plus_int(I),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4419_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_4420_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),plus_plus_rat(C,B)) ) ).

% diff_le_eq
tff(fact_4421_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_le_eq
tff(fact_4422_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_4423_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A,B)),C) ) ).

% le_diff_eq
tff(fact_4424_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A),B)),C) ) ).

% le_diff_eq
tff(fact_4425_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_4426_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_4427_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4428_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4429_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4430_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4431_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4432_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4433_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_4434_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4435_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4436_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4437_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4438_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(plus_plus_int(B),aa_int_int(minus_minus_int(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4439_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_4440_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),plus_plus_rat(C,B)) ) ).

% diff_less_eq
tff(fact_4441_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_less_eq
tff(fact_4442_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_4443_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A,B)),C) ) ).

% less_diff_eq
tff(fact_4444_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A),B)),C) ) ).

% less_diff_eq
tff(fact_4445_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_4446_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_4447_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_4448_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4449_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4450_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4451_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% zero_neq_neg_numeral
tff(fact_4452_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4453_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4454_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4455_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_4456_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Nb)) ).

% neg_numeral_le_numeral
tff(fact_4457_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_4458_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Nb)) ).

% neg_numeral_le_numeral
tff(fact_4459_neg__numeral__le__numeral,axiom,
    ! [M: 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,M))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_4460_mult__diff__mult,axiom,
    ! [Xd: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Y),aa_rat_rat(times_times_rat(A),B)) = plus_plus_rat(aa_rat_rat(times_times_rat(Xd),minus_minus_rat(Y,B)),aa_rat_rat(times_times_rat(minus_minus_rat(Xd,A)),B)) ).

% mult_diff_mult
tff(fact_4461_mult__diff__mult,axiom,
    ! [Xd: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),A)),B)) ).

% mult_diff_mult
tff(fact_4462_mult__diff__mult,axiom,
    ! [Xd: int,Y: int,A: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xd),Y)),aa_int_int(times_times_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xd),aa_int_int(minus_minus_int(Y),B))),aa_int_int(times_times_int(aa_int_int(minus_minus_int(Xd),A)),B)) ).

% mult_diff_mult
tff(fact_4463_mult__diff__mult,axiom,
    ! [Xd: code_integer,Y: code_integer,A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Y),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),minus_8373710615458151222nteger(Y,B)),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(Xd,A)),B)) ).

% mult_diff_mult
tff(fact_4464_mult__diff__mult,axiom,
    ! [Xd: complex,Y: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xd),Y)),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xd),aa_complex_complex(minus_minus_complex(Y),B))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xd),A)),B)) ).

% mult_diff_mult
tff(fact_4465_square__diff__square__factored,axiom,
    ! [Xd: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Xd),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(plus_plus_rat(Xd,Y)),minus_minus_rat(Xd,Y)) ).

% square_diff_square_factored
tff(fact_4466_square__diff__square__factored,axiom,
    ! [Xd: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd)),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,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y)) ).

% square_diff_square_factored
tff(fact_4467_square__diff__square__factored,axiom,
    ! [Xd: int,Y: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xd),Xd)),aa_int_int(times_times_int(Y),Y)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xd),Y)),aa_int_int(minus_minus_int(Xd),Y)) ).

% square_diff_square_factored
tff(fact_4468_square__diff__square__factored,axiom,
    ! [Xd: code_integer,Y: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Y)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(Xd,Y)),minus_8373710615458151222nteger(Xd,Y)) ).

% square_diff_square_factored
tff(fact_4469_square__diff__square__factored,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xd),Xd)),aa_complex_complex(times_times_complex(Y),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xd),Y)),aa_complex_complex(minus_minus_complex(Xd),Y)) ).

% square_diff_square_factored
tff(fact_4470_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( C = plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D) ) ) ).

% eq_add_iff2
tff(fact_4471_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4472_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( C = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4473_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D) )
    <=> ( C = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,A)),E),D) ) ) ).

% eq_add_iff2
tff(fact_4474_eq__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E)),D) )
    <=> ( C = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4475_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C) = D ) ) ).

% eq_add_iff1
tff(fact_4476_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4477_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4478_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D) )
    <=> ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),E),C) = D ) ) ).

% eq_add_iff1
tff(fact_4479_eq__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E)),D) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4480_neg__numeral__less__numeral,axiom,
    ! [M: 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,M))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_4481_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(Nb)) ).

% neg_numeral_less_numeral
tff(fact_4482_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_4483_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(Nb)) ).

% neg_numeral_less_numeral
tff(fact_4484_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4485_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4486_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4487_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_4488_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_4489_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_4490_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = uminus1351360451143612070nteger(A) ) ) ).

% add_eq_0_iff
tff(fact_4491_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_4492_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = zero_zero_rat )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% add_eq_0_iff
tff(fact_4493_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_4494_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_4495_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_4496_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_4497_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_4498_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_4499_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_4500_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( uminus1351360451143612070nteger(A) = B ) ) ).

% add.inverse_unique
tff(fact_4501_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_4502_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat(A,B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_4503_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4504_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4505_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = uminus1351360451143612070nteger(B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4506_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4507_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_4508_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(plus_plus_int(A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4509_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4510_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger(A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4511_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4512_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_4513_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_4514_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_4515_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != uminus1351360451143612070nteger(one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_4516_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_4517_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_4518_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_4519_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_4520_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_4521_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_4522_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_4523_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_4524_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_4525_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_4526_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_4527_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_4528_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_4529_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_4530_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_4531_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_4532_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_4533_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_4534_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_4535_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_4536_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_4537_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numeral_numeral_real(Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_4538_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != uminus1351360451143612070nteger(one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_4539_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_4540_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numeral_numeral_rat(Nb) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_4541_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_4542_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% one_neq_neg_numeral
tff(fact_4543_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_4544_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_4545_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% one_neq_neg_numeral
tff(fact_4546_numeral__times__minus__swap,axiom,
    ! [W: num,Xd: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xd)) = aa_int_int(times_times_int(Xd),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_4547_numeral__times__minus__swap,axiom,
    ! [W: num,Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_4548_numeral__times__minus__swap,axiom,
    ! [W: num,Xd: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(W)),uminus1351360451143612070nteger(Xd)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),uminus1351360451143612070nteger(numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_4549_numeral__times__minus__swap,axiom,
    ! [W: num,Xd: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xd)) = aa_complex_complex(times_times_complex(Xd),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_4550_numeral__times__minus__swap,axiom,
    ! [W: num,Xd: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(Xd)) = aa_rat_rat(times_times_rat(Xd),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_4551_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4552_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4553_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4554_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4555_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4556_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4557_square__eq__1__iff,axiom,
    ! [Xd: int] :
      ( ( aa_int_int(times_times_int(Xd),Xd) = one_one_int )
    <=> ( ( Xd = one_one_int )
        | ( Xd = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_4558_square__eq__1__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd) = one_one_real )
    <=> ( ( Xd = one_one_real )
        | ( Xd = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_4559_square__eq__1__iff,axiom,
    ! [Xd: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd) = one_one_Code_integer )
    <=> ( ( Xd = one_one_Code_integer )
        | ( Xd = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_4560_square__eq__1__iff,axiom,
    ! [Xd: complex] :
      ( ( aa_complex_complex(times_times_complex(Xd),Xd) = one_one_complex )
    <=> ( ( Xd = one_one_complex )
        | ( Xd = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_4561_square__eq__1__iff,axiom,
    ! [Xd: rat] :
      ( ( aa_rat_rat(times_times_rat(Xd),Xd) = one_one_rat )
    <=> ( ( Xd = one_one_rat )
        | ( Xd = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_4562_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != bit0(X22)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_4563_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_4564_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_4565_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> 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,M)) ) ).

% nat_less_real_le
tff(fact_4566_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_div_neg
tff(fact_4567_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_4568_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,uminus1351360451143612070nteger(B)) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_4569_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_4570_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_4571_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_neg_div
tff(fact_4572_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_4573_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(uminus1351360451143612070nteger(A),B) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_4574_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_4575_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_4576_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_4577_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(minus_minus_int(A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_4578_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_4579_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_4580_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B))) ).

% dvd_minus_mod
tff(fact_4581_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_4582_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_4583_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_4584_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_4585_Suc__to__right,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(suc,Nb) = M )
     => ( Nb = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_4586_Suc__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_4587_diff__less__Suc,axiom,
    ! [M: 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,M),Nb)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_4588_diff__less,axiom,
    ! [Nb: nat,M: 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),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),M) ) ) ).

% diff_less
tff(fact_4589_real__of__nat__div4,axiom,
    ! [Nb: nat,Xd: 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),Xd))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xd))) ).

% real_of_nat_div4
tff(fact_4590_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),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,M),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_4591_diff__add__0,axiom,
    ! [Nb: nat,M: 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),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4592_add__diff__inverse__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4593_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_4594_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_4595_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(minus_minus_int(I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_4596_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(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(minus_minus_int(I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_4597_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_4598_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),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)),M)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_4599_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_4600_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),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)),M)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_4601_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: 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)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4602_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: 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)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),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)),M) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_4603_dvd__minus__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% dvd_minus_self
tff(fact_4604_nat__minus__mod,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_4605_mod__nat__sub,axiom,
    ! [Xd: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xd),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xd),Y) ) ) ).

% mod_nat_sub
tff(fact_4606_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_4607_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ).

% mod_if
tff(fact_4608_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(times_times_int(M),Nb) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_4609_zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(times_times_int(M),Nb) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( M = 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_4610_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_4611_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_4612_nat__minus__mod__plus__right,axiom,
    ! [Nb: nat,Xd: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),M) ).

% nat_minus_mod_plus_right
tff(fact_4613_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_4614_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4615_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4616_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4617_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4618_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4619_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_4620_small__powers__of__2,axiom,
    ! [Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit1,one))),Xd)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xd),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_4621_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4622_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,A)),E),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4623_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4624_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4625_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4626_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),E),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4627_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4628_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4629_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_4630_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_4631_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_4632_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_4633_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_4634_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_4635_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_4636_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_4637_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(B,A)),E),D)) ) ).

% less_add_iff2
tff(fact_4638_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4639_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D)) ) ).

% less_add_iff2
tff(fact_4640_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4641_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),E),C)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),E),D))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(A,B)),E),C)),D) ) ).

% less_add_iff1
tff(fact_4642_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4643_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D) ) ).

% less_add_iff1
tff(fact_4644_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A),E)),C)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4645_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_4646_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_4647_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_4648_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_4649_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_4650_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_4651_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_4652_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_4653_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_4654_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_4655_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_4656_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_4657_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_4658_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_4659_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_4660_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_4661_m1mod2k,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),one_one_int) ).

% m1mod2k
tff(fact_4662_square__diff__one__factored,axiom,
    ! [Xd: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Xd),one_one_rat) = aa_rat_rat(times_times_rat(plus_plus_rat(Xd,one_one_rat)),minus_minus_rat(Xd,one_one_rat)) ).

% square_diff_one_factored
tff(fact_4663_square__diff__one__factored,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),one_one_real)) ).

% square_diff_one_factored
tff(fact_4664_square__diff__one__factored,axiom,
    ! [Xd: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xd),Xd)),one_one_int) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xd),one_one_int)),aa_int_int(minus_minus_int(Xd),one_one_int)) ).

% square_diff_one_factored
tff(fact_4665_square__diff__one__factored,axiom,
    ! [Xd: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),Xd),one_one_Code_integer) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(Xd,one_one_Code_integer)),minus_8373710615458151222nteger(Xd,one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_4666_square__diff__one__factored,axiom,
    ! [Xd: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xd),Xd)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xd),one_one_complex)),aa_complex_complex(minus_minus_complex(Xd),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4667_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xd),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4668_divide__diff__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4669_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(Xd,Z),Y) = divide_divide_rat(minus_minus_rat(Xd,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4670_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xd),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xd),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4671_diff__divide__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4672_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(Xd,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4673_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xd: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xd),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xd),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4674_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xd: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4675_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xd: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(Xd,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4676_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4677_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4678_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4679_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_4680_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_4681_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_4682_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_4683_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_4684_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_4685_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_4686_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_4687_neg__numeral__le__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_4688_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_4689_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_4690_neg__numeral__le__one,axiom,
    ! [M: 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,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_4691_neg__one__le__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_le_numeral
tff(fact_4692_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_le_numeral
tff(fact_4693_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_le_numeral
tff(fact_4694_neg__one__le__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_le_numeral
tff(fact_4695_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_4696_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_4697_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_4698_neg__numeral__le__neg__one,axiom,
    ! [M: 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,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_4699_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_4700_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_4701_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_4702_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_4703_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_le_neg_numeral
tff(fact_4704_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_one_le_neg_numeral
tff(fact_4705_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_le_neg_numeral
tff(fact_4706_not__one__le__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_le_neg_numeral
tff(fact_4707_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_4708_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_4709_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_4710_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_4711_neg__one__less__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_less_numeral
tff(fact_4712_neg__one__less__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_less_numeral
tff(fact_4713_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_less_numeral
tff(fact_4714_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_less_numeral
tff(fact_4715_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_4716_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_4717_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_4718_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_4719_not__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_less_neg_numeral
tff(fact_4720_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_less_neg_numeral
tff(fact_4721_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_one_less_neg_numeral
tff(fact_4722_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_less_neg_numeral
tff(fact_4723_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4724_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4725_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4726_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_4727_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_4728_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_4729_uminus__numeral__One,axiom,
    uminus1351360451143612070nteger(numera6620942414471956472nteger(one)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_4730_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_4731_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_4732_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_4733_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_4734_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(one))),B) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(1)
tff(fact_4735_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_4736_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_4737_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(times_times_int(B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_4738_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_4739_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),uminus1351360451143612070nteger(numera6620942414471956472nteger(one))) = uminus1351360451143612070nteger(B) ).

% mult_1s_ring_1(2)
tff(fact_4740_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_4741_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_4742_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4743_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4744_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = uminus_uminus_rat(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4745_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4746_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4747_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(divide_divide_rat(A,B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4748_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4749_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4750_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat(divide_divide_rat(A,B)) = C )
      <=> ( uminus_uminus_rat(A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4751_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_4752_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_4753_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( uminus_uminus_rat(divide_divide_rat(B,C)) = A )
    <=> $ite(C != zero_zero_rat,uminus_uminus_rat(B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_4754_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_4755_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_4756_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = uminus_uminus_rat(B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_4757_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X4: rat,K5: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,T))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K5),D3)),T)) ) ) ).

% inf_period(4)
tff(fact_4758_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X4: real,K5: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T))
        <=> ~ 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,K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4759_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int,K5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4760_inf__period_I4_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X4: code_integer,K5: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3)),T)) ) ) ).

% inf_period(4)
tff(fact_4761_inf__period_I4_J,axiom,
    ! [D: complex,D3: complex,T: complex] :
      ( aa_complex_o(dvd_dvd_complex(D),D3)
     => ! [X4: complex,K5: complex] :
          ( ~ aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(X4),T))
        <=> ~ aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4762_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X4: rat,K5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X4,T))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K5),D3)),T)) ) ) ).

% inf_period(3)
tff(fact_4763_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X4: real,K5: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T))
        <=> 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,K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4764_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int,K5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4765_inf__period_I3_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X4: code_integer,K5: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K5),D3)),T)) ) ) ).

% inf_period(3)
tff(fact_4766_inf__period_I3_J,axiom,
    ! [D: complex,D3: complex,T: complex] :
      ( aa_complex_o(dvd_dvd_complex(D),D3)
     => ! [X4: complex,K5: complex] :
          ( aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(X4),T))
        <=> aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K5),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4767_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_4768_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_4769_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_4770_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_4771_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_4772_power__minus__Bit0,axiom,
    ! [Xd: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xd)),numeral_numeral_nat(bit0(K))) = aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(K))) ).

% power_minus_Bit0
tff(fact_4773_power__minus__Bit0,axiom,
    ! [Xd: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xd)),numeral_numeral_nat(bit0(K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(K))) ).

% power_minus_Bit0
tff(fact_4774_power__minus__Bit0,axiom,
    ! [Xd: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(Xd)),numeral_numeral_nat(bit0(K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(K))) ).

% power_minus_Bit0
tff(fact_4775_power__minus__Bit0,axiom,
    ! [Xd: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),numeral_numeral_nat(bit0(K))) = aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(K))) ).

% power_minus_Bit0
tff(fact_4776_power__minus__Bit0,axiom,
    ! [Xd: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xd)),numeral_numeral_nat(bit0(K))) = aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(K))) ).

% power_minus_Bit0
tff(fact_4777_zmde,axiom,
    ! [B: int,A: int] : aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B)) = aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) ).

% zmde
tff(fact_4778_zmde,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)) = minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) ).

% zmde
tff(fact_4779_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4780_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_4781_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4782_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4783_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(B),aa_int_int(divide_divide_int(A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4784_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4785_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4786_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4787_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4788_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4789_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(minus_minus_int(A),aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_4790_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4791_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_4792_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_4793_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_rat(aa_num_num(bit1,Nb)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(Nb),numeral_numeral_rat(Nb)),one_one_rat) ).

% numeral_Bit1
tff(fact_4794_numeral__Bit1,axiom,
    ! [Nb: num] : 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,numeral_numeral_nat(Nb)),numeral_numeral_nat(Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_4795_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_4796_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : numeral_numeral_nat(aa_num_num(bit1,Nb)) = aa_nat_nat(suc,numeral_numeral_nat(bit0(Nb))) ).

% eval_nat_numeral(3)
tff(fact_4797_power__diff,axiom,
    ! [A: code_integer,Nb: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4798_power__diff,axiom,
    ! [A: complex,Nb: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4799_power__diff,axiom,
    ! [A: real,Nb: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4800_power__diff,axiom,
    ! [A: rat,Nb: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4801_power__diff,axiom,
    ! [A: nat,Nb: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4802_power__diff,axiom,
    ! [A: int,Nb: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4803_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(M))),numeral_numeral_nat(bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,Nb))),numeral_numeral_nat(bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4804_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4805_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4806_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,M))),numeral_numeral_nat(bit0(Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(bit0(Nb))),numeral_numeral_nat(bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4807_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4808_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(bit0(Nb)),numera6620942414471956472nteger(bit0(Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4809_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,M))),numeral_numeral_nat(bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,Nb))),numeral_numeral_nat(bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4810_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4811_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4812_reals__Archimedean3,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ! [Y3: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xd)) ) ).

% reals_Archimedean3
tff(fact_4813_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_4814_nz__le__conv__less,axiom,
    ! [K: nat,M: 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,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_4815_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_4816_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_4817_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_4818_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_4819_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_4820_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: 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)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),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)),M)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_4821_real__of__nat__div__aux,axiom,
    ! [Xd: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),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,Xd),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,Xd),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_4822_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)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P1,X)
          <=> aa_int_o(P1,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D))) )
       => ( ? [Z3: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P1,X) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_4823_plusinfinity,axiom,
    ! [D: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P2,X)
          <=> aa_int_o(P2,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D))) )
       => ( ? [Z3: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P2,X) ) )
         => ( ? [X_12: int] : aa_int_o(P2,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_4824_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(Nb),one_one_int)),M) ) ).

% less_1_helper
tff(fact_4825_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(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(minus_minus_int(I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_4826_int__mod__le_H,axiom,
    ! [B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(B),Nb))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(B),Nb)),aa_int_int(minus_minus_int(B),Nb)) ) ).

% int_mod_le'
tff(fact_4827_mod__nat__add,axiom,
    ! [Xd: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_4828_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,M: 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),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),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),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_4829_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B)) ) ) ).

% diff_mod_le
tff(fact_4830_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,M: 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),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4831_modulo__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_4832_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(times_times_int(aa_int_int(divide_divide_int(A),B)),B) = aa_int_int(minus_minus_int(A),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_equality_div_eq
tff(fact_4833_m1mod22k,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))),one_one_int) ).

% m1mod22k
tff(fact_4834_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( 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,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_4835_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_4836_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_4837_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,Nb: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_4838_mod__mult2__eq_H,axiom,
    ! [A: nat,M: 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,M)),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,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_4839_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xd: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4840_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xd: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xd,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4841_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xd: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4842_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xd: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xd,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xd),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4843_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4844_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4845_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4846_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4847_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4848_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4849_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4850_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4851_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4852_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4853_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_4854_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(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_4855_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B,aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4856_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4857_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B,uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4858_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4859_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4860_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C),uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4861_power2__commute,axiom,
    ! [Xd: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Xd,Y)),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Y,Xd)),numeral_numeral_nat(bit0(one))) ).

% power2_commute
tff(fact_4862_power2__commute,axiom,
    ! [Xd: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),Xd)),numeral_numeral_nat(bit0(one))) ).

% power2_commute
tff(fact_4863_power2__commute,axiom,
    ! [Xd: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xd)),numeral_numeral_nat(bit0(one))) ).

% power2_commute
tff(fact_4864_power2__commute,axiom,
    ! [Xd: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Y),Xd)),numeral_numeral_nat(bit0(one))) ).

% power2_commute
tff(fact_4865_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4866_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xd: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xd),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4867_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(Xd,Z)),Y) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(Xd),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4868_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4869_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4870_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(plus_plus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4871_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))) ) ).

% odd_mod_4_div_2
tff(fact_4872_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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,bit0(one))),A) ) ).

% even_minus
tff(fact_4873_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),uminus1351360451143612070nteger(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A) ) ).

% even_minus
tff(fact_4874_power2__eq__iff,axiom,
    ! [Xd: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))) )
    <=> ( ( Xd = Y )
        | ( Xd = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4875_power2__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))) )
    <=> ( ( Xd = Y )
        | ( Xd = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4876_power2__eq__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))) )
    <=> ( ( Xd = Y )
        | ( Xd = uminus1351360451143612070nteger(Y) ) ) ) ).

% power2_eq_iff
tff(fact_4877_power2__eq__iff,axiom,
    ! [Xd: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_complex(power_power_complex(Y),numeral_numeral_nat(bit0(one))) )
    <=> ( ( Xd = Y )
        | ( Xd = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4878_power2__eq__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))) = aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))) )
    <=> ( ( Xd = Y )
        | ( Xd = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_4879_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(aa_num_num(bit1,Nb))),numeral_numeral_nat(bit0(one))) = numeral_numeral_nat(Nb) ).

% numeral_Bit1_div_2
tff(fact_4880_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,bit0(one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_4881_real__archimedian__rdiv__eq__0,axiom,
    ! [Xd: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M2)),Xd)),C) )
         => ( Xd = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_4882_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,Nb))),numeral_numeral_nat(bit0(Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_4883_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_4884_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(bit0(Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_4885_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_4886_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_4887_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,Nb: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4888_power__diff__power__eq,axiom,
    ! [A: nat,M: 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),M)),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),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),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),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4889_power__diff__power__eq,axiom,
    ! [A: int,M: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4890_power3__eq__cube,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),A)),A) ).

% power3_eq_cube
tff(fact_4891_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),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_4892_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),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_4893_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_int_int(times_times_int(aa_int_int(times_times_int(A),A)),A) ).

% power3_eq_cube
tff(fact_4894_power3__eq__cube,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A)),A) ).

% power3_eq_cube
tff(fact_4895_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),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_4896_numeral__3__eq__3,axiom,
    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_4897_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,numeral_numeral_nat(aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_4898_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(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,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4899_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,M: 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,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4900_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_4901_add__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = $ite(M = 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,M),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_4902_div__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),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,M),Nb)),Nb)) ) ).

% div_if
tff(fact_4903_div__geq,axiom,
    ! [Nb: nat,M: 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,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_4904_le__div__geq,axiom,
    ! [Nb: nat,M: 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),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_4905_mult__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = $ite(M = 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,M),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_4906_power__sub,axiom,
    ! [Nb: nat,M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_sub
tff(fact_4907_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% nat_mult_power_less_eq
tff(fact_4908_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(minus_minus_int(X),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(minus_minus_int(X4),aa_int_int(times_times_int(K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4909_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4910_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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(divide_divide_int(B3),Nb)),aa_int_int(divide_divide_int(A2),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4911_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_4912_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(Nb),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb),aa_int_int(minus_minus_int(aa_int_int(divide_divide_int(Nb),M)),one_one_int),aa_int_int(divide_divide_int(Nb),M)) ) ) ).

% int_div_sub_1
tff(fact_4913_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_4914_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_4915_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_4916_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,bit0(one))),Nb)) ).

% of_nat_less_two_power
tff(fact_4917_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_4918_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_4919_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,plus_plus_rat(U,divide_divide_rat(aa_rat_rat(times_times_rat(R2),minus_minus_rat(V,U)),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4920_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,uminus1351360451143612070nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))))) ).

% signed_take_bit_rec
tff(fact_4921_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))))) ).

% signed_take_bit_rec
tff(fact_4922_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4923_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4924_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4925_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4926_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4927_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4928_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4929_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4930_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4931_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4932_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_4933_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(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_4934_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4935_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4936_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4937_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4938_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),numeral_numeral_nat(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_4939_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(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_4940_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_4941_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),numeral_numeral_nat(bit0(one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_4942_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_4943_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4944_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4945_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4946_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(one)),numeral_numeral_nat(bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,Nb))),numeral_numeral_nat(bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(Nb)),numeral_numeral_nat(Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4947_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4948_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4949_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(aa_num_num(bit1,M))),numeral_numeral_nat(bit0(Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(one)),numeral_numeral_nat(bit0(Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4950_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,bit0(Q2))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,bit0(Q2))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4951_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(bit0(Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(bit0(Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4952_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,numeral_numeral_nat(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_4953_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,numeral_numeral_nat(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_4954_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),uminus1351360451143612070nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_4955_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,numeral_numeral_nat(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_4956_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb),aa_nat_rat(power_power_rat(A),Nb),uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_4957_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4958_power__eq__if,axiom,
    ! [P3: assn,M: nat] :
      aa_nat_assn(power_power_assn(P3),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(times_times_assn(P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4959_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4960_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4961_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(times_times_int(P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4962_power__eq__if,axiom,
    ! [P3: code_integer,M: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P3),M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(P3),aa_nat_Code_integer(power_8256067586552552935nteger(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4963_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_4964_power__minus__mult,axiom,
    ! [Nb: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4965_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_4966_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_4967_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4968_power__minus__mult,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4969_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_4970_Suc__div__eq__add3__div,axiom,
    ! [M: 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,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),M)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_4971_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),B)) ) ) ).

% power_minus_is_div
tff(fact_4972_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => 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(power_power_nat,numeral_numeral_nat(bit0(one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_4973_Suc__mod__eq__add3__mod,axiom,
    ! [M: 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,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),M)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_4974_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,bit0(one))),aa_int_int(minus_minus_int(K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_diff_iff
tff(fact_4975_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,bit0(one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_4976_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,bit0(one))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_4977_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,bit0(one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_4978_mod__add__if__z,axiom,
    ! [Xd: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( 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),Xd)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Xd),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Xd),Y)),Z),aa_int_int(plus_plus_int(Xd),Y),aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(Xd),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_4979_mod__sub__if__z,axiom,
    ! [Xd: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( 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),Xd)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(Xd),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xd),aa_int_int(minus_minus_int(Xd),Y),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(Xd),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_4980_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4981_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4982_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4983_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4984_square__le__1,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),numeral_numeral_nat(bit0(one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_4985_square__le__1,axiom,
    ! [Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),Xd)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_4986_square__le__1,axiom,
    ! [Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_4987_square__le__1,axiom,
    ! [Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_4988_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% minus_power_mult_self
tff(fact_4989_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,numeral_numeral_nat(bit0(one))),Nb)) ).

% minus_power_mult_self
tff(fact_4990_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% minus_power_mult_self
tff(fact_4991_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,numeral_numeral_nat(bit0(one))),Nb)) ).

% minus_power_mult_self
tff(fact_4992_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% minus_power_mult_self
tff(fact_4993_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,numeral_numeral_nat(bit0(one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_4994_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,numeral_numeral_nat(bit0(one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_4995_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb),one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_4996_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,numeral_numeral_nat(bit0(one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_4997_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb),one_one_rat,uminus_uminus_rat(one_one_rat)) ).

% minus_one_power_iff
tff(fact_4998_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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,numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4999_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5000_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5001_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,bit0(one))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5002_less__two__pow__divI,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_5003_less__two__pow__divD,axiom,
    ! [Xd: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% less_two_pow_divD
tff(fact_5004_nat__power__less__diff,axiom,
    ! [Nb: nat,Q2: nat,M: 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(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ).

% nat_power_less_diff
tff(fact_5005_nat__less__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => 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(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_5006_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,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,bit0(one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5007_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,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,bit0(one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5008_power__mod__div,axiom,
    ! [Xd: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ).

% power_mod_div
tff(fact_5009_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5010_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(plus_plus_int(aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_5011_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5012_power2__diff,axiom,
    ! [Xd: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(minus_8373710615458151222nteger(Xd,Y)),numeral_numeral_nat(bit0(one))) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one)))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Xd)),Y)) ).

% power2_diff
tff(fact_5013_power2__diff,axiom,
    ! [Xd: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_complex(power_power_complex(Y),numeral_numeral_nat(bit0(one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Xd)),Y)) ).

% power2_diff
tff(fact_5014_power2__diff,axiom,
    ! [Xd: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y)),numeral_numeral_nat(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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd)),Y)) ).

% power2_diff
tff(fact_5015_power2__diff,axiom,
    ! [Xd: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xd,Y)),numeral_numeral_nat(bit0(one))) = minus_minus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Xd)),Y)) ).

% power2_diff
tff(fact_5016_power2__diff,axiom,
    ! [Xd: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xd),Y)),numeral_numeral_nat(bit0(one))) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Xd)),Y)) ).

% power2_diff
tff(fact_5017_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_5018_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_5019_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_5020_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,numeral_numeral_nat(bit0(one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5021_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,numeral_numeral_nat(bit0(one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5022_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5023_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,numeral_numeral_nat(bit0(one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5024_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_5025_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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)),Xd))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xd)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_5026_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,bit0(one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,bit0(one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_5027_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_5028_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5029_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5030_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B)))
         => ( aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5031_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5032_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5033_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5034_exp__div__exp__eq,axiom,
    ! [M: 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,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ).

% exp_div_exp_eq
tff(fact_5035_exp__div__exp__eq,axiom,
    ! [M: nat,Nb: nat] :
      aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),M) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ).

% exp_div_exp_eq
tff(fact_5036_exp__div__exp__eq,axiom,
    ! [M: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),M) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ).

% exp_div_exp_eq
tff(fact_5037_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(bit0(one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))) ) ).

% even_mod_4_div_2
tff(fact_5038_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,bit0(one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5039_sb__dec__lem_H,axiom,
    ! [K: 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,bit0(one))),K)),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),A)) ) ).

% sb_dec_lem'
tff(fact_5040_space__cnt,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T))) ).

% space_cnt
tff(fact_5041_t__build__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_5042_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_5043_vebt__buildup__bound,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_5044_linear__plus__1__le__power,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => 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)),Xd)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_5045_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(bit0(one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(bit0(one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(bit0(one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5046_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_5047_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_5048_buildup__build__time,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb)) ).

% buildup_build_time
tff(fact_5049_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_5050_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_5051_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = numeral_numeral_nat(V) ) ) ).

% int_eq_iff_numeral
tff(fact_5052_negative__eq__positive,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Nb = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_5053_real__add__minus__iff,axiom,
    ! [Xd: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xd = A ) ) ).

% real_add_minus_iff
tff(fact_5054_int__dvd__int__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% int_dvd_int_iff
tff(fact_5055_negative__zless,axiom,
    ! [Nb: nat,M: 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,M)) ).

% negative_zless
tff(fact_5056_minus__real__def,axiom,
    ! [Xd: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_5057_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A4: real,B2: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A4),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A4),C3) ) ) ) )
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A4: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A4),X)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A4)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A4),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_5058_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,numeral_numeral_nat(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_5059_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_5060_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_5061_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_5062_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_5063_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_5064_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_5065_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_5066_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_5067_zadd__int__left,axiom,
    ! [M: nat,Nb: nat,Z: int] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))),Z) ).

% zadd_int_left
tff(fact_5068_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_5069_int__plus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_5070_not__int__zless__negative,axiom,
    ! [Nb: nat,M: 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,M))) ).

% not_int_zless_negative
tff(fact_5071_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_5072_real__minus__mult__self__le,axiom,
    ! [U: real,Xd: 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,Xd),Xd)) ).

% real_minus_mult_self_le
tff(fact_5073_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_5074_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_5075_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_5076_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_5077_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_5078_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_5079_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Nb = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_5080_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_5081_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_5082_real__add__less__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xd)) ) ).

% real_add_less_0_iff
tff(fact_5083_real__0__less__add__iff,axiom,
    ! [Xd: 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,Xd),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xd)),Y) ) ).

% real_0_less_add_iff
tff(fact_5084_real__add__le__0__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xd)) ) ).

% real_add_le_0_iff
tff(fact_5085_real__0__le__add__iff,axiom,
    ! [Xd: 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,Xd),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xd)),Y) ) ).

% real_0_le_add_iff
tff(fact_5086_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_5087_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_5088_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_5089_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(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_5090_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_5091_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_5092_negD,axiom,
    ! [Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),zero_zero_int)
     => ? [N: nat] : Xd = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_5093_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_5094_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_5095_realpow__square__minus__le,axiom,
    ! [U: real,Xd: 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),numeral_numeral_nat(bit0(one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))) ).

% realpow_square_minus_le
tff(fact_5096_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xd: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xd),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xd)
         => aa_int_o(P,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Xd)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_5097_real__of__nat__div2,axiom,
    ! [Nb: nat,Xd: 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,Xd))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xd)))) ).

% real_of_nat_div2
tff(fact_5098_real__of__nat__div3,axiom,
    ! [Nb: nat,Xd: 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,Xd))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xd)))),one_one_real) ).

% real_of_nat_div3
tff(fact_5099_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = numeral_numeral_nat(aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_5100_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = numeral_numeral_nat(bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_5101_Bernoulli__inequality,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => 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)),Xd))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xd)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_5102_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_5103_reals__Archimedean2,axiom,
    ! [Xd: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_5104_reals__Archimedean2,axiom,
    ! [Xd: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_5105_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_5106_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_5107_ex__less__of__nat__mult,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xd)) ) ).

% ex_less_of_nat_mult
tff(fact_5108_ex__less__of__nat__mult,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xd)) ) ).

% ex_less_of_nat_mult
tff(fact_5109_space__space_H,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)) ).

% space_space'
tff(fact_5110_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(bit0(one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_second
tff(fact_5111_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(bit0(one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_first
tff(fact_5112_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_5113_cnt__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one)))))))) ) ).

% cnt_bound
tff(fact_5114_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)),one_one_real))) ) ).

% cnt_bound'
tff(fact_5115_space_H__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_5116_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),zero_zero_nat) ).

% valid_0_not
tff(fact_5117_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_5118_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_5119_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_5120_space__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_5121_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),A,B)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_if
tff(fact_5122_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_5123_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_5124_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xd: nat,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xd)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ) ).

% member_bound
tff(fact_5125_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_5126_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5127_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5128_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_5129_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5130_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_5131_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xd)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xd)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xd) ) ) ) ).

% member_valid_both_member_options
tff(fact_5132_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_5133_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_5134_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_5135_member__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd)
      <=> aa_set_nat_o(member_nat(Xd),vEBT_set_vebt(T)) ) ) ).

% member_correct
tff(fact_5136_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5137_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = numeral_numeral_nat(K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_5138_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( numeral_numeral_nat(K) = aa_nat_nat(suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_5139_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa_num_num(bit1,K)) = numeral_numeral_nat(bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_5140_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_5141_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_5142_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)),numeral_numeral_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_5143_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_5144_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_5145_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,pred_numeral(K)),Nb) ).

% diff_numeral_Suc
tff(fact_5146_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_5147_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_5148_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_5149_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_5150_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_5151_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_bit0
tff(fact_5152_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_5153_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_5154_numeral__eq__Suc,axiom,
    ! [K: num] : numeral_numeral_nat(K) = aa_nat_nat(suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_5155_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(K)),one_one_nat) ).

% pred_numeral_def
tff(fact_5156_dbl__dec__def,axiom,
    ! [Xd: rat] : neg_nu3179335615603231917ec_rat(Xd) = minus_minus_rat(plus_plus_rat(Xd,Xd),one_one_rat) ).

% dbl_dec_def
tff(fact_5157_dbl__dec__def,axiom,
    ! [Xd: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xd) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xd),Xd)),one_one_complex) ).

% dbl_dec_def
tff(fact_5158_dbl__dec__def,axiom,
    ! [Xd: real] : aa_real_real(neg_nu6075765906172075777c_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Xd)),one_one_real) ).

% dbl_dec_def
tff(fact_5159_dbl__dec__def,axiom,
    ! [Xd: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xd) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(Xd),Xd)),one_one_int) ).

% dbl_dec_def
tff(fact_5160_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),numeral_numeral_nat(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one))))))))) ) ).

% delete_bound_height
tff(fact_5161_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% delete_bound_height'
tff(fact_5162_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_insert(T,Xd)),Y)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
              | ( Xd = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_5163_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Deg) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Deg)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_5164_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5165_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),numeral_numeral_nat(bit0(one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5166_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_5167_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5168_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5169_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_5170_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),Nb)) ).

% pochhammer_double
tff(fact_5171_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),Nb)) ).

% pochhammer_double
tff(fact_5172_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),Nb)) ).

% pochhammer_double
tff(fact_5173_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_5174_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_5175_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_5176_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_5177_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5178_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_5179_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(bit0(one)))))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_5180_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_5181_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_5182_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_5183_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_5184_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_5185_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_5186_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_5187_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_5188_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5189_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_5190_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5191_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5192_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_5193_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_5194_log__less__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( 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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_5195_log__less__one__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_5196_one__less__log__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xd) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_5197_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_5198_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_5199_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_5200_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K))) ).

% dbl_inc_simps(1)
tff(fact_5201_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K))) = uminus1351360451143612070nteger(neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_5202_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_5203_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K))) ).

% dbl_inc_simps(1)
tff(fact_5204_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_5205_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K))) ).

% dbl_dec_simps(1)
tff(fact_5206_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K))) = uminus1351360451143612070nteger(neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_5207_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_5208_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu3179335615603231917ec_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K))) ).

% dbl_dec_simps(1)
tff(fact_5209_log__le__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( 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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_5210_log__le__one__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_5211_one__le__log__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xd) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_5212_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_5213_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_5214_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_5215_pochhammer__pos,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => 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,Xd),Nb)) ) ).

% pochhammer_pos
tff(fact_5216_pochhammer__pos,axiom,
    ! [Xd: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xd),Nb)) ) ).

% pochhammer_pos
tff(fact_5217_pochhammer__pos,axiom,
    ! [Xd: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
     => 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,Xd),Nb)) ) ).

% pochhammer_pos
tff(fact_5218_pochhammer__pos,axiom,
    ! [Xd: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xd)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xd),Nb)) ) ).

% pochhammer_pos
tff(fact_5219_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,M: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5220_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,M: 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),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5221_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,M: 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),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5222_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5223_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5224_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5225_log__base__change,axiom,
    ! [A: real,B: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_5226_log__of__power__eq,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_5227_less__log__of__power,axiom,
    ! [B: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% less_log_of_power
tff(fact_5228_pochhammer__nonneg,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => 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,Xd),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5229_pochhammer__nonneg,axiom,
    ! [Xd: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xd),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5230_pochhammer__nonneg,axiom,
    ! [Xd: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xd)
     => 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,Xd),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5231_pochhammer__nonneg,axiom,
    ! [Xd: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xd)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xd),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5232_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_5233_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_5234_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_5235_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_5236_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_5237_log__mult,axiom,
    ! [A: real,Xd: 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),Xd)
         => ( 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,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_5238_le__log__of__power,axiom,
    ! [B: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M)) ) ) ).

% le_log_of_power
tff(fact_5239_log__divide,axiom,
    ! [A: real,Xd: 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),Xd)
         => ( 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,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_5240_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xd: 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)),Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_5241_log__nat__power,axiom,
    ! [Xd: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ).

% log_nat_power
tff(fact_5242_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(A,one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_5243_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_5244_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_5245_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_5246_pochhammer__rec,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer)),Nb)) ).

% pochhammer_rec
tff(fact_5247_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_5248_dbl__inc__def,axiom,
    ! [Xd: real] : aa_real_real(neg_nu8295874005876285629c_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Xd)),one_one_real) ).

% dbl_inc_def
tff(fact_5249_dbl__inc__def,axiom,
    ! [Xd: rat] : neg_nu5219082963157363817nc_rat(Xd) = plus_plus_rat(plus_plus_rat(Xd,Xd),one_one_rat) ).

% dbl_inc_def
tff(fact_5250_dbl__inc__def,axiom,
    ! [Xd: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xd) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(Xd),Xd)),one_one_int) ).

% dbl_inc_def
tff(fact_5251_log2__of__power__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_5252_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_5253_pochhammer__rec_H,axiom,
    ! [Z: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)) ).

% pochhammer_rec'
tff(fact_5254_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_5255_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_5256_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_5257_pochhammer__rec_H,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)) ).

% pochhammer_rec'
tff(fact_5258_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),plus_plus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_5259_pochhammer__Suc,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb)),plus_p5714425477246183910nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) ).

% pochhammer_Suc
tff(fact_5260_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_5261_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(plus_plus_int(A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_5262_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_5263_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_5264_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_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5265_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(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_5266_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_5267_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_5268_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_5269_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(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_5270_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_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_5271_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_5272_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_5273_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_5274_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_5275_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_5276_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_5277_log__of__power__less,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_5278_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,Nb))),M)) ).

% pochhammer_product'
tff(fact_5279_pochhammer__product_H,axiom,
    ! [Z: code_integer,Nb: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),M)) ).

% pochhammer_product'
tff(fact_5280_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),M)) ).

% pochhammer_product'
tff(fact_5281_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_int_int(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),M)) ).

% pochhammer_product'
tff(fact_5282_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),M)) ).

% pochhammer_product'
tff(fact_5283_pochhammer__product_H,axiom,
    ! [Z: complex,Nb: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))),M)) ).

% pochhammer_product'
tff(fact_5284_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_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5285_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(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_5286_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_5287_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_5288_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_5289_log__of__power__le,axiom,
    ! [M: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_5290_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5291_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5292_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5293_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(times_times_int(aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5294_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5295_pochhammer__product,axiom,
    ! [M: nat,Nb: nat,Z: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% pochhammer_product
tff(fact_5296_less__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_5297_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_5298_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(R2)),K)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(uminus1351360451143612070nteger(R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_5299_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(R2,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(uminus_uminus_rat(R2),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_5300_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_5301_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(times_times_int(aa_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(times_times_int(R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_5302_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_5303_log2__of__power__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_5304_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_5305_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(B)),K)) ).

% pochhammer_minus'
tff(fact_5306_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K)) ).

% pochhammer_minus'
tff(fact_5307_pochhammer__minus_H,axiom,
    ! [B: 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,B),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,B)),K)) ).

% pochhammer_minus'
tff(fact_5308_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_5309_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_5310_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(B)),K) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_5311_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_5312_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(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,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_5313_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_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(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_5314_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_5315_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5316_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_5317_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5318_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_5319_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_5320_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_5321_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_5322_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_5323_norm__le__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xd)),zero_zero_real)
    <=> ( Xd = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5324_norm__le__zero__iff,axiom,
    ! [Xd: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),zero_zero_real)
    <=> ( Xd = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5325_zero__less__norm__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xd))
    <=> ( Xd != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_5326_zero__less__norm__iff,axiom,
    ! [Xd: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xd))
    <=> ( Xd != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_5327_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_5328_norm__eq__zero,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_5329_norm__eq__zero,axiom,
    ! [Xd: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_5330_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_5331_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_5332_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_5333_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_5334_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5335_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_5336_pred__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xd),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xd)
        & ! [Z5: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Z5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),Xd) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z5),Y) ) ) ) ).

% pred_member
tff(fact_5337_succ__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xd),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y)
        & ! [Z5: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Z5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Z5) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z5) ) ) ) ).

% succ_member
tff(fact_5338_norm__not__less__zero,axiom,
    ! [Xd: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_5339_norm__mult,axiom,
    ! [Xd: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xd)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_5340_norm__mult,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_5341_norm__ge__zero,axiom,
    ! [Xd: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xd)) ).

% norm_ge_zero
tff(fact_5342_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_5343_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_5344_norm__power,axiom,
    ! [Xd: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xd)),Nb) ).

% norm_power
tff(fact_5345_norm__power,axiom,
    ! [Xd: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xd),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),Nb) ).

% norm_power
tff(fact_5346_norm__uminus__minus,axiom,
    ! [Xd: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xd)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) ).

% norm_uminus_minus
tff(fact_5347_norm__uminus__minus,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xd)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xd),Y)) ).

% norm_uminus_minus
tff(fact_5348_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5349_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5350_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5351_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_5352_norm__mult__less,axiom,
    ! [Xd: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xd)),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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_5353_norm__mult__less,axiom,
    ! [Xd: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),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(Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_5354_norm__mult__ineq,axiom,
    ! [Xd: 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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xd)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_5355_norm__mult__ineq,axiom,
    ! [Xd: 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(Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_mult_ineq
tff(fact_5356_norm__add__less,axiom,
    ! [Xd: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xd)),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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_5357_norm__add__less,axiom,
    ! [Xd: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),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(Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_5358_norm__triangle__lt,axiom,
    ! [Xd: 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,Xd)),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,Xd),Y))),E) ) ).

% norm_triangle_lt
tff(fact_5359_norm__triangle__lt,axiom,
    ! [Xd: 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,Xd)),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(Xd),Y))),E) ) ).

% norm_triangle_lt
tff(fact_5360_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: 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,B)),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),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_5361_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: 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,B)),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),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_5362_norm__triangle__ineq,axiom,
    ! [Xd: 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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xd)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_5363_norm__triangle__ineq,axiom,
    ! [Xd: 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(Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_5364_norm__triangle__le,axiom,
    ! [Xd: 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,Xd)),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,Xd),Y))),E) ) ).

% norm_triangle_le
tff(fact_5365_norm__triangle__le,axiom,
    ! [Xd: 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,Xd)),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(Xd),Y))),E) ) ).

% norm_triangle_le
tff(fact_5366_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_5367_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_5368_norm__diff__triangle__less,axiom,
    ! [Xd: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5369_norm__diff__triangle__less,axiom,
    ! [Xd: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xd),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xd),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5370_norm__power__ineq,axiom,
    ! [Xd: 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,Xd),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xd)),Nb)) ).

% norm_power_ineq
tff(fact_5371_norm__power__ineq,axiom,
    ! [Xd: 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(Xd),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),Nb)) ).

% norm_power_ineq
tff(fact_5372_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_5373_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_5374_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_5375_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_5376_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5377_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_5378_square__norm__one,axiom,
    ! [Xd: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xd) = one_one_real ) ) ).

% square_norm_one
tff(fact_5379_square__norm__one,axiom,
    ! [Xd: complex] :
      ( ( aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xd) = one_one_real ) ) ).

% square_norm_one
tff(fact_5380_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5381_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5382_inrange,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_5383_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_5384_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_5385_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Xd)),Xd) ) ) ).

% valid_insert_both_member_options_add
tff(fact_5386_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xd)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),Xd) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_5387_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5388_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xd)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd) ) ) ).

% valid_member_both_member_options
tff(fact_5389_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xd)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd) ) ) ).

% both_member_options_equiv_member
tff(fact_5390_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xd)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,T),Xd)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,T),Xd) ) ) ).

% both_member_options_def
tff(fact_5391_ceiling__zero,axiom,
    archim2889992004027027881ng_rat(zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_5392_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_5393_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_5394_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_5395_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_5396_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_5397_ceiling__le__zero,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xd)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_5398_ceiling__le__zero,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xd)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_5399_ceiling__le__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_5400_ceiling__le__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_5401_zero__less__ceiling,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd) ) ).

% zero_less_ceiling
tff(fact_5402_zero__less__ceiling,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% zero_less_ceiling
tff(fact_5403_numeral__less__ceiling,axiom,
    ! [V: num,Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(V)),Xd) ) ).

% numeral_less_ceiling
tff(fact_5404_numeral__less__ceiling,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(V)),Xd) ) ).

% numeral_less_ceiling
tff(fact_5405_ceiling__less__one,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_5406_ceiling__less__one,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_5407_one__le__ceiling,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd) ) ).

% one_le_ceiling
tff(fact_5408_one__le__ceiling,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% one_le_ceiling
tff(fact_5409_ceiling__add__numeral,axiom,
    ! [Xd: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),numeral_numeral_real(V))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xd)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5410_ceiling__add__numeral,axiom,
    ! [Xd: rat,V: num] : archim2889992004027027881ng_rat(plus_plus_rat(Xd,numeral_numeral_rat(V))) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(Xd)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5411_ceiling__le__one,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xd)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real) ) ).

% ceiling_le_one
tff(fact_5412_ceiling__le__one,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xd)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),one_one_rat) ) ).

% ceiling_le_one
tff(fact_5413_one__less__ceiling,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xd) ) ).

% one_less_ceiling
tff(fact_5414_one__less__ceiling,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd) ) ).

% one_less_ceiling
tff(fact_5415_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_5416_ceiling__neg__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_5417_ceiling__diff__numeral,axiom,
    ! [Xd: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),numeral_numeral_real(V))) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xd)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5418_ceiling__diff__numeral,axiom,
    ! [Xd: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(Xd,numeral_numeral_rat(V))) = aa_int_int(minus_minus_int(archim2889992004027027881ng_rat(Xd)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5419_ceiling__add__one,axiom,
    ! [Xd: rat] : archim2889992004027027881ng_rat(plus_plus_rat(Xd,one_one_rat)) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(Xd)),one_one_int) ).

% ceiling_add_one
tff(fact_5420_ceiling__add__one,axiom,
    ! [Xd: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xd)),one_one_int) ).

% ceiling_add_one
tff(fact_5421_ceiling__numeral__power,axiom,
    ! [Xd: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ).

% ceiling_numeral_power
tff(fact_5422_ceiling__numeral__power,axiom,
    ! [Xd: num,Nb: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ).

% ceiling_numeral_power
tff(fact_5423_ceiling__diff__one,axiom,
    ! [Xd: rat] : archim2889992004027027881ng_rat(minus_minus_rat(Xd,one_one_rat)) = aa_int_int(minus_minus_int(archim2889992004027027881ng_rat(Xd)),one_one_int) ).

% ceiling_diff_one
tff(fact_5424_ceiling__diff__one,axiom,
    ! [Xd: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),one_one_real)) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xd)),one_one_int) ).

% ceiling_diff_one
tff(fact_5425_ceiling__less__zero,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_5426_ceiling__less__zero,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),uminus_uminus_rat(one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_5427_zero__le__ceiling,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd) ) ).

% zero_le_ceiling
tff(fact_5428_zero__le__ceiling,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),Xd) ) ).

% zero_le_ceiling
tff(fact_5429_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_5430_ceiling__less__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_5431_ceiling__less__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_5432_numeral__le__ceiling,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)),Xd) ) ).

% numeral_le_ceiling
tff(fact_5433_numeral__le__ceiling,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),Xd) ) ).

% numeral_le_ceiling
tff(fact_5434_ceiling__le__neg__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xd)),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,Xd),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5435_ceiling__le__neg__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xd)),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,Xd),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5436_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xd) ) ).

% neg_numeral_less_ceiling
tff(fact_5437_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),Xd) ) ).

% neg_numeral_less_ceiling
tff(fact_5438_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_5439_ceiling__less__neg__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),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,Xd),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_5440_ceiling__less__neg__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),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,Xd),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_5441_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xd) ) ).

% neg_numeral_le_ceiling
tff(fact_5442_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),Xd) ) ).

% neg_numeral_le_ceiling
tff(fact_5443_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5444_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5445_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(ord_less_eq_num(A),B)
          | ( aa_num_o(ord_less_eq_num(C),A)
            & aa_num_o(ord_less_eq_num(B),D)
            & ( aa_num_o(ord_less_num(C),A)
              | aa_num_o(ord_less_num(B),D) ) ) )
        & aa_num_o(ord_less_eq_num(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5446_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5447_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5448_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C,D))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5449_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_5450_ceiling__less__cancel,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),archim2889992004027027881ng_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y) ) ).

% ceiling_less_cancel
tff(fact_5451_ceiling__less__cancel,axiom,
    ! [Xd: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% ceiling_less_cancel
tff(fact_5452_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
          & aa_nat_o(P,M4) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X5) ) ) ).

% ex_nat_less
tff(fact_5453_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),Nb)
         => aa_nat_o(P,M4) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X5) ) ) ).

% all_nat_less
tff(fact_5454_ceiling__add__le,axiom,
    ! [Xd: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(plus_plus_rat(Xd,Y))),aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(Xd)),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_5455_ceiling__add__le,axiom,
    ! [Xd: 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,Xd),Y))),aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xd)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_5456_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(times_times_int(aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_5457_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B))),aa_int_int(times_times_int(archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).

% mult_ceiling_le
tff(fact_5458_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_5459_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(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)),numeral_numeral_nat(bit0(one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_5460_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5461_artanh__def,axiom,
    ! [Xd: real] : aa_real_real(artanh_real,Xd) = 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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xd)))),numeral_numeral_real(bit0(one))) ).

% artanh_def
tff(fact_5462_log__ceil__idem,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),Xd)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xd)))) ) ) ).

% log_ceil_idem
tff(fact_5463_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5464_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xd),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),X5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X5) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_5465_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xd),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xd)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xd)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_5466_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_5467_ln__less__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5468_ln__inj__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xd) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xd = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5469_of__int__ceiling__cancel,axiom,
    ! [Xd: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xd)) = Xd )
    <=> ? [N6: int] : Xd = aa_int_real(ring_1_of_int_real,N6) ) ).

% of_int_ceiling_cancel
tff(fact_5470_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5471_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5472_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5473_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5474_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5475_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5476_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5477_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5478_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5479_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5480_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5481_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5482_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_5483_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_5484_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_5485_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_5486_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5487_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5488_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5489_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5490_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5491_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5492_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5493_ln__le__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5494_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5495_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5496_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5497_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5498_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5499_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5500_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5501_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5502_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(W),Z)) = plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5503_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(plus_plus_int(W),Z)) = aa_int_int(plus_plus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_5504_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_5505_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(plus_plus_int(W),Z)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_add
tff(fact_5506_ln__less__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xd)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_5507_ln__gt__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xd))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd) ) ) ).

% ln_gt_zero_iff
tff(fact_5508_ln__eq__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( ( aa_real_real(ln_ln_real,Xd) = zero_zero_real )
      <=> ( Xd = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_5509_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(times_times_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_5510_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(times_times_int(W),Z)) = aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_5511_of__int__mult,axiom,
    ! [W: int,Z: int] : ring_18347121197199848620nteger(aa_int_int(times_times_int(W),Z)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(ring_18347121197199848620nteger(W)),ring_18347121197199848620nteger(Z)) ).

% of_int_mult
tff(fact_5512_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(times_times_int(W),Z)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W)),aa_int_complex(ring_17405671764205052669omplex,Z)) ).

% of_int_mult
tff(fact_5513_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5514_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_5515_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_5516_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_5517_of__int__power,axiom,
    ! [Z: int,Nb: nat] : ring_18347121197199848620nteger(aa_nat_int(power_power_int(Z),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(Z)),Nb) ).

% of_int_power
tff(fact_5518_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,Xd) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xd ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5519_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,Xd) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xd ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5520_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,Xd) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xd ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5521_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W) = ring_18347121197199848620nteger(Xd) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xd ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5522_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xd) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xd = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5523_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xd) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xd = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5524_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xd) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( Xd = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5525_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( ( ring_18347121197199848620nteger(Xd) = aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W) )
    <=> ( Xd = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5526_ln__le__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xd)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_5527_ln__ge__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xd))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ) ).

% ln_ge_zero_iff
tff(fact_5528_ceiling__add__of__int,axiom,
    ! [Xd: rat,Z: int] : archim2889992004027027881ng_rat(plus_plus_rat(Xd,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(plus_plus_int(archim2889992004027027881ng_rat(Xd)),Z) ).

% ceiling_add_of_int
tff(fact_5529_ceiling__add__of__int,axiom,
    ! [Xd: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xd)),Z) ).

% ceiling_add_of_int
tff(fact_5530_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5531_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5532_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5533_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5534_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5535_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5536_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5537_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5538_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5539_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5540_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5541_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5542_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5543_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5544_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5545_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5546_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5547_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5548_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5549_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5550_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5551_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5552_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5553_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5554_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5555_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5556_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5557_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5558_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5559_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5560_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5561_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5562_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5563_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5564_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5565_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5566_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb) = ring_18347121197199848620nteger(Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5567_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xd)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5568_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5569_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5570_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5571_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5572_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5573_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5574_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5575_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5576_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5577_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5578_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xd)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5579_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xd)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5580_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5581_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W)),ring_18347121197199848620nteger(Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5582_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5583_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5584_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5585_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5586_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xd)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5587_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xd: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xd)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5588_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(ring_18347121197199848620nteger(B)),W)),ring_18347121197199848620nteger(Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5589_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5590_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5591_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xd) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5592_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb)),ring_18347121197199848620nteger(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,Xd)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5593_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5594_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5595_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: 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,Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5596_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),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,Xd)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5597_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),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,Xd)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5598_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),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,Xd)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5599_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,Xd)),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,Xd)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5600_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),Nb)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5601_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5602_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5603_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: 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,Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5604_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xd)),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,Xd)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5605_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),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,Xd)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5606_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),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,Xd)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5607_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,Xd)),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,Xd)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5608_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),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,Xd))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5609_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),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,Xd))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5610_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),Nb) = ring_18347121197199848620nteger(Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5611_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xd))),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,Xd))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5612_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),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,Xd))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5613_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: 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,Xd))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5614_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5615_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Y) = aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5616_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xd))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5617_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xd))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5618_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),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,Xd))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5619_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),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,Xd))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5620_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),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,Xd))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5621_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,Xd))),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,Xd))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5622_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5623_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5624_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5625_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xd: 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,Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5626_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: 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,Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5627_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5628_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5629_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),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,Xd))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5630_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,Xd))),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,Xd))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5631_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xd))),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,Xd))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5632_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xd))),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,Xd))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5633_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xd))),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,Xd))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5634_ex__less__of__int,axiom,
    ! [Xd: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_5635_ex__less__of__int,axiom,
    ! [Xd: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_5636_ex__of__int__less,axiom,
    ! [Xd: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xd) ).

% ex_of_int_less
tff(fact_5637_ex__of__int__less,axiom,
    ! [Xd: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xd) ).

% ex_of_int_less
tff(fact_5638_mult__of__int__commute,axiom,
    ! [Xd: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xd)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xd)) ).

% mult_of_int_commute
tff(fact_5639_mult__of__int__commute,axiom,
    ! [Xd: int,Y: int] : aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,Xd)),Y) = aa_int_int(times_times_int(Y),aa_int_int(ring_1_of_int_int,Xd)) ).

% mult_of_int_commute
tff(fact_5640_mult__of__int__commute,axiom,
    ! [Xd: int,Y: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(ring_18347121197199848620nteger(Xd)),Y) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),ring_18347121197199848620nteger(Xd)) ).

% mult_of_int_commute
tff(fact_5641_mult__of__int__commute,axiom,
    ! [Xd: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Xd)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,Xd)) ).

% mult_of_int_commute
tff(fact_5642_ln__less__self,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xd)),Xd) ) ).

% ln_less_self
tff(fact_5643_log__def,axiom,
    ! [A: real,Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xd)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_5644_ln__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xd)),Xd) ) ).

% ln_bound
tff(fact_5645_ln__gt__zero__imp__gt__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xd))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5646_ln__less__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xd)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5647_ln__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xd)) ) ).

% ln_gt_zero
tff(fact_5648_ln__ge__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xd)) ) ).

% ln_ge_zero
tff(fact_5649_bset_I1_J,axiom,
    ! [D3: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B3)
                 => ( X != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(minus_minus_int(X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B3)
                   => ( X != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(minus_minus_int(X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(minus_minus_int(X4),D3))
                & aa_int_o(Q,aa_int_int(minus_minus_int(X4),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_5650_bset_I2_J,axiom,
    ! [D3: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B3)
                 => ( X != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(minus_minus_int(X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B3)
                   => ( X != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(minus_minus_int(X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(minus_minus_int(X4),D3))
                | aa_int_o(Q,aa_int_int(minus_minus_int(X4),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_5651_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(plus_plus_int(X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(plus_plus_int(X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(plus_plus_int(X4),D3))
                & aa_int_o(Q,aa_int_int(plus_plus_int(X4),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_5652_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A2)
                 => ( X != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(plus_plus_int(X),D3)) ) )
     => ( ! [X: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A2)
                   => ( X != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X)
             => aa_int_o(Q,aa_int_int(plus_plus_int(X),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(plus_plus_int(X4),D3))
                | aa_int_o(Q,aa_int_int(plus_plus_int(X4),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_5653_ceiling__le,axiom,
    ! [Xd: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),A) ) ).

% ceiling_le
tff(fact_5654_ceiling__le,axiom,
    ! [Xd: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xd)),A) ) ).

% ceiling_le
tff(fact_5655_less__ceiling__iff,axiom,
    ! [Z: int,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xd) ) ).

% less_ceiling_iff
tff(fact_5656_less__ceiling__iff,axiom,
    ! [Z: int,Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xd) ) ).

% less_ceiling_iff
tff(fact_5657_real__of__int__div4,axiom,
    ! [Nb: int,Xd: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xd))),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,Xd))) ).

% real_of_int_div4
tff(fact_5658_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_5659_ln__ge__zero__imp__ge__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xd))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5660_ln__add__one__self__le__self,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => 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),Xd))),Xd) ) ).

% ln_add_one_self_le_self
tff(fact_5661_ln__mult,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xd)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_5662_ln__eq__minus__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( ( aa_real_real(ln_ln_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),one_one_real) )
       => ( Xd = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5663_ln__div,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xd)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_5664_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_5665_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_5666_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_5667_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_5668_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_5669_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_5670_floor__exists1,axiom,
    ! [Xd: real] :
    ? [X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),Xd)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(X),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),Xd)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Y3),one_one_int))) )
         => ( Y3 = X ) ) ) ).

% floor_exists1
tff(fact_5671_floor__exists1,axiom,
    ! [Xd: rat] :
    ? [X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),Xd)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(X),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),Xd)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Y3),one_one_int))) )
         => ( Y3 = X ) ) ) ).

% floor_exists1
tff(fact_5672_floor__exists,axiom,
    ! [Xd: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xd)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Z2),one_one_int))) ) ).

% floor_exists
tff(fact_5673_floor__exists,axiom,
    ! [Xd: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xd)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Z2),one_one_int))) ) ).

% floor_exists
tff(fact_5674_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_5675_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,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_5676_of__int__neg__numeral,axiom,
    ! [K: num] : ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_5677_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_5678_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))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_5679_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_5680_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2))),plus_plus_rat(R2,one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_5681_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_5682_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2)),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_5683_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xd: 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,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xd) ) ).

% of_nat_less_of_int_iff
tff(fact_5684_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xd: 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,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xd) ) ).

% of_nat_less_of_int_iff
tff(fact_5685_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xd: 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,Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xd) ) ).

% of_nat_less_of_int_iff
tff(fact_5686_aset_I10_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D3)),T)) ) ) ) ).

% aset(10)
tff(fact_5687_aset_I9_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D3)),T)) ) ) ) ).

% aset(9)
tff(fact_5688_bset_I10_J,axiom,
    ! [D: int,D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B3)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D3)),T)) ) ) ) ).

% bset(10)
tff(fact_5689_bset_I9_J,axiom,
    ! [D: int,D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B3)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(X4),T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D3)),T)) ) ) ) ).

% bset(9)
tff(fact_5690_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
    <=> 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,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_5691_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),M)
    <=> 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,M)) ) ).

% int_less_real_le
tff(fact_5692_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_5693_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_5694_real__of__int__div__aux,axiom,
    ! [Xd: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xd)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Xd),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(Xd),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5695_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(bit0(one)))),one_one_real) ).

% ln_2_less_1
tff(fact_5696_ln__le__minus__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_5697_ln__diff__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( 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,Xd)),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,Xd),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_5698_ln__add__one__self__le__self2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => 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),Xd))),Xd) ) ).

% ln_add_one_self_le_self2
tff(fact_5699_ln__realpow,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xd)) ) ) ).

% ln_realpow
tff(fact_5700_ceiling__correct,axiom,
    ! [Xd: 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,Xd))),one_one_real)),Xd)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xd))) ) ).

% ceiling_correct
tff(fact_5701_ceiling__correct,axiom,
    ! [Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xd)),one_one_rat)),Xd)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xd))) ) ).

% ceiling_correct
tff(fact_5702_ceiling__unique,axiom,
    ! [Z: int,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xd) = Z ) ) ) ).

% ceiling_unique
tff(fact_5703_ceiling__unique,axiom,
    ! [Z: int,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(Xd) = Z ) ) ) ).

% ceiling_unique
tff(fact_5704_ceiling__eq__iff,axiom,
    ! [Xd: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xd) = 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)),Xd)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_5705_ceiling__eq__iff,axiom,
    ! [Xd: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(Xd) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),Xd)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_5706_ceiling__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,T))
    <=> ! [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)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_5707_ceiling__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,archim2889992004027027881ng_rat(T))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2),one_one_rat)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_5708_ceiling__less__iff,axiom,
    ! [Xd: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xd)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_5709_ceiling__less__iff,axiom,
    ! [Xd: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xd)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_5710_le__ceiling__iff,axiom,
    ! [Z: int,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xd) ) ).

% le_ceiling_iff
tff(fact_5711_le__ceiling__iff,axiom,
    ! [Z: int,Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xd) ) ).

% le_ceiling_iff
tff(fact_5712_bset_I3_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(minus_minus_int(X4),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5713_bset_I4_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(minus_minus_int(X4),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5714_bset_I5_J,axiom,
    ! [D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B3)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(X4),D3)),T) ) ) ) ).

% bset(5)
tff(fact_5715_bset_I7_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(minus_minus_int(X4),D3)) ) ) ) ) ).

% bset(7)
tff(fact_5716_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X4 = T )
             => ( aa_int_int(plus_plus_int(X4),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5717_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X4 != T )
             => ( aa_int_int(plus_plus_int(X4),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5718_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(T),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(X4),D3)),T) ) ) ) ) ).

% aset(5)
tff(fact_5719_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(plus_plus_int(X4),D3)) ) ) ) ).

% aset(7)
tff(fact_5720_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)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P,X)
          <=> aa_int_o(P,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D))) )
       => ( ? [X_13: int] : aa_int_o(P,X_13)
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X5) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5721_real__of__int__div2,axiom,
    ! [Nb: int,Xd: 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,Xd))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xd)))) ).

% real_of_int_div2
tff(fact_5722_real__of__int__div3,axiom,
    ! [Nb: int,Xd: 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,Xd))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xd)))),one_one_real) ).

% real_of_int_div3
tff(fact_5723_ln__one__minus__pos__upper__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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),Xd))),aa_real_real(uminus_uminus_real,Xd)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_5724_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_5725_ceiling__divide__upper,axiom,
    ! [Q2: real,P3: 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,P3),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,P3),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_5726_ceiling__divide__upper,axiom,
    ! [Q2: rat,P3: 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,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_5727_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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,bit0(one))),K) ) ).

% even_of_int_iff
tff(fact_5728_bset_I6_J,axiom,
    ! [D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B3)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(X4),D3)),T) ) ) ) ).

% bset(6)
tff(fact_5729_bset_I8_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(T),one_one_int)),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B3)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(minus_minus_int(X4),D3)) ) ) ) ) ).

% bset(8)
tff(fact_5730_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(T),one_one_int)),A2)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A2)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(X4),D3)),T) ) ) ) ) ).

% aset(6)
tff(fact_5731_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A2)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(plus_plus_int(X4),D3)) ) ) ) ).

% aset(8)
tff(fact_5732_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P2: fun_int_o,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),B3)
                     => ( X != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(minus_minus_int(X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P2,X5) )
                | ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B3)
                        & aa_int_o(P,aa_int_int(plus_plus_int(Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5733_cppi,axiom,
    ! [D3: int,P: fun_int_o,P2: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),A2)
                     => ( X != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(plus_plus_int(X),D3)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(minus_minus_int(X),aa_int_int(times_times_int(K2),D3))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P2,X5) )
                | ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A2)
                        & aa_int_o(P,aa_int_int(minus_minus_int(Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5734_ceiling__divide__lower,axiom,
    ! [Q2: real,P3: 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,P3),Q2)))),one_one_real)),Q2)),P3) ) ).

% ceiling_divide_lower
tff(fact_5735_ceiling__divide__lower,axiom,
    ! [Q2: rat,P3: 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(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P3,Q2))),one_one_rat)),Q2)),P3) ) ).

% ceiling_divide_lower
tff(fact_5736_ceiling__eq,axiom,
    ! [Nb: int,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd) = aa_int_int(plus_plus_int(Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5737_ceiling__eq,axiom,
    ! [Nb: int,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Nb),one_one_rat))
       => ( archim2889992004027027881ng_rat(Xd) = aa_int_int(plus_plus_int(Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5738_ln__one__plus__pos__lower__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xd))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5739_member__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_5740_ln__one__minus__pos__lower__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xd))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5741_psubsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( ( A2 != B3 )
       => aa_set_nat_o(ord_less_set_nat(A2),B3) ) ) ).

% psubsetI
tff(fact_5742_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_5743_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_5744_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_5745_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,one))))))) ) ).

% succ_bound_height
tff(fact_5746_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),numeral_numeral_nat(aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% pred_bound_height
tff(fact_5747_psubsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B3) ) ) ).

% psubsetD
tff(fact_5748_psubsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A2),B3)
     => ( aa_set_real_o(member_real(C),A2)
       => aa_set_real_o(member_real(C),B3) ) ) ).

% psubsetD
tff(fact_5749_psubsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A2),B3)
     => ( aa_set_int_o(member_int(C),A2)
       => aa_set_int_o(member_int(C),B3) ) ) ).

% psubsetD
tff(fact_5750_psubsetD,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B3)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A2)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B3) ) ) ).

% psubsetD
tff(fact_5751_psubsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B3)
     => ( aa_set_complex_o(member_complex(C),A2)
       => aa_set_complex_o(member_complex(C),B3) ) ) ).

% psubsetD
tff(fact_5752_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B3: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B3)
     => ? [B2: real] : aa_set_real_o(member_real(B2),minus_minus_set_real(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5753_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B3: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B3)
     => ? [B2: int] : aa_set_int_o(member_int(B2),minus_minus_set_int(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5754_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B3)
     => ? [B2: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5755_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B3)
     => ? [B2: complex] : aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5756_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => ? [B2: nat] : aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_5757_psubsetE,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
         => aa_set_nat_o(ord_less_eq_set_nat(B3),A2) ) ) ).

% psubsetE
tff(fact_5758_psubset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
        & ( A2 != B3 ) ) ) ).

% psubset_eq
tff(fact_5759_psubset__imp__subset,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B3) ) ).

% psubset_imp_subset
tff(fact_5760_psubset__subset__trans,axiom,
    ! [A2: set_nat,B3: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C5)
       => aa_set_nat_o(ord_less_set_nat(A2),C5) ) ) ).

% psubset_subset_trans
tff(fact_5761_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B3),A2) ) ) ).

% subset_not_subset_eq
tff(fact_5762_subset__psubset__trans,axiom,
    ! [A2: set_nat,B3: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(ord_less_set_nat(B3),C5)
       => aa_set_nat_o(ord_less_set_nat(A2),C5) ) ) ).

% subset_psubset_trans
tff(fact_5763_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B3)
        | ( A2 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_5764_insersimp,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Y)),numeral_numeral_nat(aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_5765_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(one))))))) ) ).

% insert_bound_height
tff(fact_5766_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c2(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_5767_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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),Xd))),Xd))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5768_round__unique,axiom,
    ! [Xd: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xd) = Y ) ) ) ).

% round_unique
tff(fact_5769_round__unique,axiom,
    ! [Xd: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(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)),plus_plus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))))
       => ( archim7778729529865785530nd_rat(Xd) = Y ) ) ) ).

% round_unique
tff(fact_5770_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,U: real,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d2(T),Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_5771_tanh__ln__real,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xd)) = 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,Xd),numeral_numeral_nat(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,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5772_obtain__set__pred,axiom,
    ! [Z: nat,Xd: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xd)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A2),Xd),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_5773_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_5774_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)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X4) ) ) ) ).

% succ_none_empty
tff(fact_5775_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),A) ) ) ) ).

% pred_none_empty
tff(fact_5776_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_5777_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_5778_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_5779_abs__abs,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_abs
tff(fact_5780_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_5781_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_5782_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5783_abs__idempotent,axiom,
    ! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).

% abs_idempotent
tff(fact_5784_obtain__set__succ,axiom,
    ! [Xd: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,B3)
         => ( ( A2 = B3 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A2),Xd),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_5785_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5786_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5787_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5788_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5789_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5790_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_5791_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_5792_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_5793_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5794_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_5795_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_5796_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_5797_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5798_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_5799_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5800_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5801_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5802_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_5803_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,numeral_numeral_real(Nb)) = numeral_numeral_real(Nb) ).

% abs_numeral
tff(fact_5804_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_rat(numeral_numeral_rat(Nb)) = numeral_numeral_rat(Nb) ).

% abs_numeral
tff(fact_5805_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_5806_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_5807_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_5808_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B))) = plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_5809_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_5810_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5811_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5812_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5813_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5814_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_5815_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_5816_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(times_times_int(A),A) ).

% abs_mult_self_eq
tff(fact_5817_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),A) ).

% abs_mult_self_eq
tff(fact_5818_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_5819_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_5820_abs__divide,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_divide
tff(fact_5821_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_5822_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_5823_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_5824_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_5825_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_5826_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_5827_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_5828_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_5829_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_5830_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% dvd_abs_iff
tff(fact_5831_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_5832_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_5833_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),abs_abs_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_5834_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% abs_dvd_iff
tff(fact_5835_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_5836_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_5837_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,abs_abs_rat(M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_5838_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_5839_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_5840_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_5841_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_5842_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,aa_o_real(zero_n3304061248610475627l_real,(P))) = aa_o_real(zero_n3304061248610475627l_real,(P)) ).

% abs_bool_eq
tff(fact_5843_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_rat(zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_5844_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_5845_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_5846_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_5847_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5848_tanh__real__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(tanh_real,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_5849_tanh__real__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xd)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% tanh_real_less_iff
tff(fact_5850_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_5851_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_5852_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5853_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_5854_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_5855_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_5856_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_5857_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_5858_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_5859_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_5860_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5861_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_5862_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_5863_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_5864_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),abs_abs_rat(A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5865_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_5866_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_5867_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(Nb) ).

% abs_neg_numeral
tff(fact_5868_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_5869_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(Nb) ).

% abs_neg_numeral
tff(fact_5870_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_5871_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_5872_abs__neg__one,axiom,
    abs_abs_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5873_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_5874_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_5875_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_5876_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_5877_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_5878_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_5879_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_5880_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5881_round__0,axiom,
    archim7778729529865785530nd_rat(zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5882_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5883_round__numeral,axiom,
    ! [Nb: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5884_tanh__real__pos__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% tanh_real_pos_iff
tff(fact_5885_tanh__real__neg__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_5886_tanh__real__nonpos__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_5887_tanh__real__nonneg__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% tanh_real_nonneg_iff
tff(fact_5888_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5889_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_5890_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5891_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5892_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5893_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5894_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_5895_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_nonpos
tff(fact_5896_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_5897_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_5898_artanh__minus__real,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xd)) ) ) ).

% artanh_minus_real
tff(fact_5899_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5900_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_5901_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5902_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_5903_abs__power2,axiom,
    ! [A: rat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one)))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one))) ).

% abs_power2
tff(fact_5904_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one))) ).

% abs_power2
tff(fact_5905_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one)))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) ).

% abs_power2
tff(fact_5906_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) ).

% abs_power2
tff(fact_5907_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(abs_abs_rat(A)),numeral_numeral_nat(bit0(one))) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(bit0(one))) ).

% power2_abs
tff(fact_5908_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),numeral_numeral_nat(bit0(one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one))) ).

% power2_abs
tff(fact_5909_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),numeral_numeral_nat(bit0(one))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) ).

% power2_abs
tff(fact_5910_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),numeral_numeral_nat(bit0(one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(bit0(one))) ).

% power2_abs
tff(fact_5911_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_5912_round__neg__numeral,axiom,
    ! [Nb: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_5913_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),numeral_numeral_nat(W)) = aa_nat_rat(power_power_rat(A),numeral_numeral_nat(W)) ) ) ).

% power_even_abs_numeral
tff(fact_5914_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),numeral_numeral_nat(W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(W)) ) ) ).

% power_even_abs_numeral
tff(fact_5915_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),numeral_numeral_nat(W)) = aa_nat_int(power_power_int(A),numeral_numeral_nat(W)) ) ) ).

% power_even_abs_numeral
tff(fact_5916_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),numeral_numeral_nat(W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),numeral_numeral_nat(W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),numeral_numeral_nat(W)) ) ) ).

% power_even_abs_numeral
tff(fact_5917_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5918_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_5919_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_5920_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5921_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_5922_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_5923_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_5924_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_5925_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_5926_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_5927_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_5928_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),abs_abs_rat(A)) ).

% abs_ge_self
tff(fact_5929_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_5930_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5931_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5932_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5933_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5934_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_5935_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_5936_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_5937_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_5938_abs__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_mult
tff(fact_5939_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_5940_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).

% abs_minus_commute
tff(fact_5941_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_5942_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B)) = aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A)) ).

% abs_minus_commute
tff(fact_5943_abs__eq__iff,axiom,
    ! [Xd: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xd) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xd = Y )
        | ( Xd = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5944_abs__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xd) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xd = Y )
        | ( Xd = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5945_abs__eq__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(Xd) = abs_abs_Code_integer(Y) )
    <=> ( ( Xd = Y )
        | ( Xd = uminus1351360451143612070nteger(Y) ) ) ) ).

% abs_eq_iff
tff(fact_5946_abs__eq__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( ( abs_abs_rat(Xd) = abs_abs_rat(Y) )
    <=> ( ( Xd = Y )
        | ( Xd = uminus_uminus_rat(Y) ) ) ) ).

% abs_eq_iff
tff(fact_5947_power__abs,axiom,
    ! [A: rat,Nb: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb) ).

% power_abs
tff(fact_5948_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_5949_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_5950_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_5951_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_5952_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_5953_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_5954_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( abs_abs_rat(L) = abs_abs_rat(K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5955_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M4: nat] :
        ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),M4) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_5956_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Nb) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_5957_finite__set__decode,axiom,
    ! [Nb: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(nat_set_decode,Nb)) ).

% finite_set_decode
tff(fact_5958_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_5959_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_5960_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A)) ).

% abs_ge_zero
tff(fact_5961_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_5962_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_5963_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_5964_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_5965_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_5966_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_5967_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_5968_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_5969_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_5970_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_5971_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_5972_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(plus_plus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B))) ).

% abs_triangle_ineq
tff(fact_5973_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_5974_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(C),D)) ) ) ).

% abs_mult_less
tff(fact_5975_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_5976_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_5977_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(times_times_int(C),D)) ) ) ).

% abs_mult_less
tff(fact_5978_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5979_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5980_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5981_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5982_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_5983_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_5984_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq3
tff(fact_5985_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B))) ).

% abs_triangle_ineq3
tff(fact_5986_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_5987_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_5988_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq2
tff(fact_5989_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B))) ).

% abs_triangle_ineq2
tff(fact_5990_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_5991_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% nonzero_abs_divide
tff(fact_5992_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_5993_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_5994_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A)) ).

% abs_ge_minus_self
tff(fact_5995_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_5996_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_5997_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B) ) ) ).

% abs_le_iff
tff(fact_5998_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_le_iff
tff(fact_5999_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_6000_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_6001_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B) ) ).

% abs_le_D2
tff(fact_6002_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ).

% abs_le_D2
tff(fact_6003_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_6004_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_6005_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_6006_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B) ) ) ).

% abs_leI
tff(fact_6007_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_6008_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_6009_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_6010_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A)),B) ) ) ).

% abs_less_iff
tff(fact_6011_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_less_iff
tff(fact_6012_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_6013_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_6014_tanh__real__lt__1,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xd)),one_one_real) ).

% tanh_real_lt_1
tff(fact_6015_dense__eq0__I,axiom,
    ! [Xd: 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,Xd)),E2) )
     => ( Xd = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6016_dense__eq0__I,axiom,
    ! [Xd: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xd)),E2) )
     => ( Xd = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6017_abs__mult__pos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xd) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xd)) ) ) ).

% abs_mult_pos
tff(fact_6018_abs__mult__pos,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xd)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(Y)),Xd) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Y),Xd)) ) ) ).

% abs_mult_pos
tff(fact_6019_abs__mult__pos,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
     => ( aa_rat_rat(times_times_rat(abs_abs_rat(Y)),Xd) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),Xd)) ) ) ).

% abs_mult_pos
tff(fact_6020_abs__mult__pos,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_int(times_times_int(aa_int_int(abs_abs_int,Y)),Xd) = aa_int_int(abs_abs_int,aa_int_int(times_times_int(Y),Xd)) ) ) ).

% abs_mult_pos
tff(fact_6021_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6022_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6023_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_6024_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6025_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_6026_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6027_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6028_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_6029_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6030_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = uminus1351360451143612070nteger(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6031_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat(A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = uminus_uminus_rat(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6032_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6033_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6034_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = uminus1351360451143612070nteger(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6035_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = uminus_uminus_rat(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6036_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6037_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_6038_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6039_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6040_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_6041_abs__div__pos,axiom,
    ! [Y: real,Xd: 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,Xd)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) ) ) ).

% abs_div_pos
tff(fact_6042_abs__div__pos,axiom,
    ! [Y: rat,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( divide_divide_rat(abs_abs_rat(Xd),Y) = abs_abs_rat(divide_divide_rat(Xd,Y)) ) ) ).

% abs_div_pos
tff(fact_6043_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_6044_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_6045_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),uminus1351360451143612070nteger(X4),X4) ).

% abs_if_raw
tff(fact_6046_abs__if__raw,axiom,
    ! [X4: rat] :
      abs_abs_rat(X4) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),zero_zero_rat),uminus_uminus_rat(X4),X4) ).

% abs_if_raw
tff(fact_6047_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_6048_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_6049_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = uminus1351360451143612070nteger(A) ) ) ).

% abs_of_neg
tff(fact_6050_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_6051_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_6052_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_6053_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A),A) ).

% abs_if
tff(fact_6054_abs__if,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),uminus_uminus_rat(A),A) ).

% abs_if
tff(fact_6055_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6056_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6057_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(plus_plus_rat(A,B),plus_plus_rat(C,D)))),plus_plus_rat(abs_abs_rat(minus_minus_rat(A,C)),abs_abs_rat(minus_minus_rat(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6058_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A),B)),aa_int_int(plus_plus_int(C),D)))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),C))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6059_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6060_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6061_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B))) ).

% abs_triangle_ineq4
tff(fact_6062_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A),B))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6063_abs__diff__le__iff,axiom,
    ! [Xd: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xd,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xd)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xd),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6064_abs__diff__le__iff,axiom,
    ! [Xd: 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,Xd),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)),Xd)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6065_abs__diff__le__iff,axiom,
    ! [Xd: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Xd,A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,R2)),Xd)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),plus_plus_rat(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6066_abs__diff__le__iff,axiom,
    ! [Xd: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xd),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A),R2)),Xd)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),aa_int_int(plus_plus_int(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6067_abs__diff__less__iff,axiom,
    ! [Xd: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xd,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xd)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xd),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6068_abs__diff__less__iff,axiom,
    ! [Xd: 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,Xd),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xd)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6069_abs__diff__less__iff,axiom,
    ! [Xd: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xd,A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,R2)),Xd)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6070_abs__diff__less__iff,axiom,
    ! [Xd: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xd),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A),R2)),Xd)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_int_int(plus_plus_int(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6071_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_6072_lemma__interval__lt,axiom,
    ! [A: real,Xd: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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,Xd),Y3))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_6073_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_6074_tanh__real__gt__neg1,axiom,
    ! [Xd: 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,Xd)) ).

% tanh_real_gt_neg1
tff(fact_6075_abs__add__one__gt__zero,axiom,
    ! [Xd: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xd))) ).

% abs_add_one_gt_zero
tff(fact_6076_abs__add__one__gt__zero,axiom,
    ! [Xd: 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,Xd))) ).

% abs_add_one_gt_zero
tff(fact_6077_abs__add__one__gt__zero,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,abs_abs_rat(Xd))) ).

% abs_add_one_gt_zero
tff(fact_6078_abs__add__one__gt__zero,axiom,
    ! [Xd: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),aa_int_int(abs_abs_int,Xd))) ).

% abs_add_one_gt_zero
tff(fact_6079_of__int__leD,axiom,
    ! [Nb: int,Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xd) ) ) ).

% of_int_leD
tff(fact_6080_of__int__leD,axiom,
    ! [Nb: int,Xd: 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))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ) ).

% of_int_leD
tff(fact_6081_of__int__leD,axiom,
    ! [Nb: int,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Nb))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xd) ) ) ).

% of_int_leD
tff(fact_6082_of__int__leD,axiom,
    ! [Nb: int,Xd: 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))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xd) ) ) ).

% of_int_leD
tff(fact_6083_of__int__lessD,axiom,
    ! [Nb: int,Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xd) ) ) ).

% of_int_lessD
tff(fact_6084_of__int__lessD,axiom,
    ! [Nb: int,Xd: 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))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd) ) ) ).

% of_int_lessD
tff(fact_6085_of__int__lessD,axiom,
    ! [Nb: int,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Nb))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xd) ) ) ).

% of_int_lessD
tff(fact_6086_of__int__lessD,axiom,
    ! [Nb: int,Xd: 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))),Xd)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xd) ) ) ).

% of_int_lessD
tff(fact_6087_lemma__interval,axiom,
    ! [A: real,Xd: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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,Xd),Y3))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_6088_of__int__round__abs__le,axiom,
    ! [Xd: 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,Xd))),Xd))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))) ).

% of_int_round_abs_le
tff(fact_6089_of__int__round__abs__le,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xd)),Xd))),divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))) ).

% of_int_round_abs_le
tff(fact_6090_round__unique_H,axiom,
    ! [Xd: 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,Xd),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xd) = Nb ) ) ).

% round_unique'
tff(fact_6091_round__unique_H,axiom,
    ! [Xd: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xd,aa_int_rat(ring_1_of_int_rat,Nb)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))
     => ( archim7778729529865785530nd_rat(Xd) = Nb ) ) ).

% round_unique'
tff(fact_6092_abs__le__square__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_6093_abs__le__square__iff,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xd)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_6094_abs__le__square__iff,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xd)),abs_abs_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_6095_abs__le__square__iff,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xd)),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(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one)))) ) ).

% abs_le_square_iff
tff(fact_6096_abs__square__eq__1,axiom,
    ! [Xd: rat] :
      ( ( aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one))) = one_one_rat )
    <=> ( abs_abs_rat(Xd) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6097_abs__square__eq__1,axiom,
    ! [Xd: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xd) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6098_abs__square__eq__1,axiom,
    ! [Xd: int] :
      ( ( aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xd) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6099_abs__square__eq__1,axiom,
    ! [Xd: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xd) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6100_power__even__abs,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_even_abs
tff(fact_6101_power__even__abs,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_6102_power__even__abs,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_6103_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_6104_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6105_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),numeral_numeral_nat(bit0(one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xd)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6106_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xd: 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(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_rat(power_power_rat(Y),numeral_numeral_nat(bit0(one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xd)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6107_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xd: 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(Xd),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xd)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6108_abs__square__le__1,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real) ) ).

% abs_square_le_1
tff(fact_6109_abs__square__le__1,axiom,
    ! [Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xd)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6110_abs__square__le__1,axiom,
    ! [Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xd)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6111_abs__square__le__1,axiom,
    ! [Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xd)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6112_abs__square__less__1,axiom,
    ! [Xd: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xd)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6113_abs__square__less__1,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),one_one_real) ) ).

% abs_square_less_1
tff(fact_6114_abs__square__less__1,axiom,
    ! [Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(Xd)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6115_abs__square__less__1,axiom,
    ! [Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xd)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6116_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_6117_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6118_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),abs_abs_rat(B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6119_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6120_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% pred_bound_height'
tff(fact_6121_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c2(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% succ'_bound_height
tff(fact_6122_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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),Xd))),Xd))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6123_of__int__round__le,axiom,
    ! [Xd: 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,Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))) ).

% of_int_round_le
tff(fact_6124_of__int__round__le,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xd))),plus_plus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))) ).

% of_int_round_le
tff(fact_6125_of__int__round__ge,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xd))) ).

% of_int_round_ge
tff(fact_6126_of__int__round__ge,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xd))) ).

% of_int_round_ge
tff(fact_6127_of__int__round__gt,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xd))) ).

% of_int_round_gt
tff(fact_6128_of__int__round__gt,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xd))) ).

% of_int_round_gt
tff(fact_6129_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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),Xd))),Xd))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6130_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xd: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => aa_real_o(aa_real_fun_real_o(P,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_6131_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xd: code_integer] :
      ( ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),numeral_numeral_nat(bit0(one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xd)),aa_nat_Code_integer(power_8256067586552552935nteger(Xd),numeral_numeral_nat(bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_6132_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xd: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
         => aa_rat_o(aa_rat_fun_rat_o(P,X),aa_nat_rat(power_power_rat(X),numeral_numeral_nat(bit0(one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,abs_abs_rat(Xd)),aa_nat_rat(power_power_rat(Xd),numeral_numeral_nat(bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_6133_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xd: int] :
      ( ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_int_o(aa_int_fun_int_o(P,X),aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xd)),aa_nat_int(power_power_int(Xd),numeral_numeral_nat(bit0(one)))) ) ).

% abs_sqrt_wlog
tff(fact_6134_arctan__double,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,Xd)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))))) ) ) ).

% arctan_double
tff(fact_6135_ceiling__log__eq__powr__iff,axiom,
    ! [Xd: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),Xd)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_6136_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6137_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary))) ).

% height_compose_summary
tff(fact_6138_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(bit0(one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(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,numeral_numeral_nat(bit0(one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_6139_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Nb)
     => ( Deg = Nb ) ) ).

% deg_deg_n
tff(fact_6140_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList2),S2) ) ).

% deg_SUcn_Node
tff(fact_6141_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_6142_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_6143_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z) = zero_zero_real )
    <=> ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_6144_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_6145_zdvd1__eq,axiom,
    ! [Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xd),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xd) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6146_of__int__floor__cancel,axiom,
    ! [Xd: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xd)) = Xd )
    <=> ? [N6: int] : Xd = aa_int_real(ring_1_of_int_real,N6) ) ).

% of_int_floor_cancel
tff(fact_6147_arctan__eq__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(arctan,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_6148_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_6149_powr__zero__eq__one,axiom,
    ! [Xd: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),zero_zero_real) = $ite(Xd = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_6150_floor__zero,axiom,
    archim3151403230148437115or_rat(zero_zero_rat) = zero_zero_int ).

% floor_zero
tff(fact_6151_floor__zero,axiom,
    aa_real_int(archim6058952711729229775r_real,zero_zero_real) = zero_zero_int ).

% floor_zero
tff(fact_6152_floor__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_6153_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_6154_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_6155_floor__one,axiom,
    archim3151403230148437115or_rat(one_one_rat) = one_one_int ).

% floor_one
tff(fact_6156_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_6157_powr__gt__zero,axiom,
    ! [Xd: 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,Xd),A))
    <=> ( Xd != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_6158_powr__nonneg__iff,axiom,
    ! [A: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xd)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_6159_powr__less__cancel__iff,axiom,
    ! [Xd: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_6160_arctan__less__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_6161_zero__less__arctan__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% zero_less_arctan_iff
tff(fact_6162_arctan__le__zero__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_6163_zero__le__arctan__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% zero_le_arctan_iff
tff(fact_6164_powr__eq__one__iff,axiom,
    ! [A: real,Xd: 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),Xd) = one_one_real )
      <=> ( Xd = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_6165_powr__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),one_one_real) = Xd ) ) ).

% powr_one
tff(fact_6166_powr__one__gt__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),one_one_real) = Xd )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% powr_one_gt_zero_iff
tff(fact_6167_powr__le__cancel__iff,axiom,
    ! [Xd: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_6168_numeral__powr__numeral__real,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),numeral_numeral_nat(Nb)) ).

% numeral_powr_numeral_real
tff(fact_6169_zero__le__floor,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd) ) ).

% zero_le_floor
tff(fact_6170_zero__le__floor,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% zero_le_floor
tff(fact_6171_numeral__le__floor,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(V)),Xd) ) ).

% numeral_le_floor
tff(fact_6172_numeral__le__floor,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(V)),Xd) ) ).

% numeral_le_floor
tff(fact_6173_floor__less__zero,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_6174_floor__less__zero,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% floor_less_zero
tff(fact_6175_floor__less__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),numeral_numeral_rat(V)) ) ).

% floor_less_numeral
tff(fact_6176_floor__less__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),numeral_numeral_real(V)) ) ).

% floor_less_numeral
tff(fact_6177_zero__less__floor,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xd) ) ).

% zero_less_floor
tff(fact_6178_zero__less__floor,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ).

% zero_less_floor
tff(fact_6179_floor__le__zero,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xd)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),one_one_rat) ) ).

% floor_le_zero
tff(fact_6180_floor__le__zero,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xd)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ).

% floor_le_zero
tff(fact_6181_one__le__floor,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xd) ) ).

% one_le_floor
tff(fact_6182_one__le__floor,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd) ) ).

% one_le_floor
tff(fact_6183_floor__less__one,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),one_one_rat) ) ).

% floor_less_one
tff(fact_6184_floor__less__one,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ).

% floor_less_one
tff(fact_6185_floor__neg__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_6186_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_6187_floor__diff__numeral,axiom,
    ! [Xd: rat,V: num] : archim3151403230148437115or_rat(minus_minus_rat(Xd,numeral_numeral_rat(V))) = aa_int_int(minus_minus_int(archim3151403230148437115or_rat(Xd)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_6188_floor__diff__numeral,axiom,
    ! [Xd: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),numeral_numeral_real(V))) = aa_int_int(minus_minus_int(aa_real_int(archim6058952711729229775r_real,Xd)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_6189_floor__numeral__power,axiom,
    ! [Xd: num,Nb: nat] : archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xd)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ).

% floor_numeral_power
tff(fact_6190_floor__numeral__power,axiom,
    ! [Xd: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xd)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ).

% floor_numeral_power
tff(fact_6191_floor__diff__one,axiom,
    ! [Xd: rat] : archim3151403230148437115or_rat(minus_minus_rat(Xd,one_one_rat)) = aa_int_int(minus_minus_int(archim3151403230148437115or_rat(Xd)),one_one_int) ).

% floor_diff_one
tff(fact_6192_floor__diff__one,axiom,
    ! [Xd: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),one_one_real)) = aa_int_int(minus_minus_int(aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int) ).

% floor_diff_one
tff(fact_6193_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_6194_powr__log__cancel,axiom,
    ! [A: real,Xd: 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),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)) = Xd ) ) ) ) ).

% powr_log_cancel
tff(fact_6195_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_6196_powr__numeral,axiom,
    ! [Xd: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(Nb)) ) ) ).

% powr_numeral
tff(fact_6197_numeral__less__floor,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(numeral_numeral_rat(V),one_one_rat)),Xd) ) ).

% numeral_less_floor
tff(fact_6198_numeral__less__floor,axiom,
    ! [V: num,Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)),Xd) ) ).

% numeral_less_floor
tff(fact_6199_floor__le__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_6200_floor__le__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xd)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_6201_one__less__floor,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(bit0(one))),Xd) ) ).

% one_less_floor
tff(fact_6202_one__less__floor,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),Xd) ) ).

% one_less_floor
tff(fact_6203_floor__le__one,axiom,
    ! [Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xd)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),numeral_numeral_rat(bit0(one))) ) ).

% floor_le_one
tff(fact_6204_floor__le__one,axiom,
    ! [Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),numeral_numeral_real(bit0(one))) ) ).

% floor_le_one
tff(fact_6205_neg__numeral__le__floor,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(V))),Xd) ) ).

% neg_numeral_le_floor
tff(fact_6206_neg__numeral__le__floor,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xd) ) ).

% neg_numeral_le_floor
tff(fact_6207_floor__less__neg__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% floor_less_neg_numeral
tff(fact_6208_floor__less__neg__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% floor_less_neg_numeral
tff(fact_6209_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_6210_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_6211_neg__numeral__less__floor,axiom,
    ! [V: num,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),Xd) ) ).

% neg_numeral_less_floor
tff(fact_6212_neg__numeral__less__floor,axiom,
    ! [V: num,Xd: 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,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xd) ) ).

% neg_numeral_less_floor
tff(fact_6213_floor__le__neg__numeral,axiom,
    ! [Xd: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xd)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_6214_floor__le__neg__numeral,axiom,
    ! [Xd: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xd)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_6215_square__powr__half,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(abs_abs_real,Xd) ).

% square_powr_half
tff(fact_6216_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_6217_powr__powr,axiom,
    ! [Xd: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_6218_arctan__monotone,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xd)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_6219_arctan__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xd)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% arctan_less_iff
tff(fact_6220_floor__less__cancel,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),archim3151403230148437115or_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y) ) ).

% floor_less_cancel
tff(fact_6221_floor__less__cancel,axiom,
    ! [Xd: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% floor_less_cancel
tff(fact_6222_powr__non__neg,axiom,
    ! [A: real,Xd: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xd)),zero_zero_real) ).

% powr_non_neg
tff(fact_6223_powr__less__mono2__neg,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_6224_powr__ge__pzero,axiom,
    ! [Xd: 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,Xd),Y)) ).

% powr_ge_pzero
tff(fact_6225_powr__mono2,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mono2
tff(fact_6226_powr__less__mono,axiom,
    ! [A: real,B: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B)) ) ) ).

% powr_less_mono
tff(fact_6227_powr__less__cancel,axiom,
    ! [Xd: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_6228_powr__mono,axiom,
    ! [A: real,B: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B)) ) ) ).

% powr_mono
tff(fact_6229_abs__zmult__eq__1,axiom,
    ! [M: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(M),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6230_abs__div,axiom,
    ! [Y: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xd)
     => ( aa_int_int(abs_abs_int,aa_int_int(divide_divide_int(Xd),Y)) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,Xd)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_6231_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_6232_floor__less__iff,axiom,
    ! [Xd: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xd)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_6233_floor__less__iff,axiom,
    ! [Xd: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xd)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_6234_floor__add__int,axiom,
    ! [Xd: rat,Z: int] : aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),Z) = archim3151403230148437115or_rat(plus_plus_rat(Xd,aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_6235_floor__add__int,axiom,
    ! [Xd: real,Z: int] : aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_6236_int__add__floor,axiom,
    ! [Z: int,Xd: rat] : aa_int_int(plus_plus_int(Z),archim3151403230148437115or_rat(Xd)) = archim3151403230148437115or_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),Xd)) ).

% int_add_floor
tff(fact_6237_int__add__floor,axiom,
    ! [Z: int,Xd: real] : aa_int_int(plus_plus_int(Z),aa_real_int(archim6058952711729229775r_real,Xd)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),Xd)) ).

% int_add_floor
tff(fact_6238_le__floor__add,axiom,
    ! [Xd: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),archim3151403230148437115or_rat(Y))),archim3151403230148437115or_rat(plus_plus_rat(Xd,Y))) ).

% le_floor_add
tff(fact_6239_le__floor__add,axiom,
    ! [Xd: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y))) ).

% le_floor_add
tff(fact_6240_powr__mono2_H,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),A)) ) ) ) ).

% powr_mono2'
tff(fact_6241_powr__less__mono2,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_6242_floor__power,axiom,
    ! [Xd: real,Nb: nat] :
      ( ( Xd = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xd)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,Xd)),Nb) ) ) ).

% floor_power
tff(fact_6243_powr__inj,axiom,
    ! [A: real,Xd: 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),Xd) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( Xd = Y ) ) ) ) ).

% powr_inj
tff(fact_6244_gr__one__powr,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => ( 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,Xd),Y)) ) ) ).

% gr_one_powr
tff(fact_6245_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : archim3151403230148437115or_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(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_6246_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(divide_divide_int(K),L) ).

% floor_divide_of_int_eq
tff(fact_6247_powr__le1,axiom,
    ! [A: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_6248_powr__mono__both,axiom,
    ! [A: real,B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_6249_ge__one__powr__ge__zero,axiom,
    ! [Xd: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => ( 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,Xd),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_6250_powr__divide,axiom,
    ! [Xd: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_6251_powr__mult,axiom,
    ! [Xd: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_6252_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_6253_log__base__powr,axiom,
    ! [A: real,B: real,Xd: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)),B) ) ) ).

% log_base_powr
tff(fact_6254_log__powr,axiom,
    ! [Xd: real,B: real,Y: real] :
      ( ( Xd != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ).

% log_powr
tff(fact_6255_ln__powr,axiom,
    ! [Xd: real,Y: real] :
      ( ( Xd != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xd)) ) ) ).

% ln_powr
tff(fact_6256_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_6257_powr__add,axiom,
    ! [Xd: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),B)) ).

% powr_add
tff(fact_6258_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6259_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_6260_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_6261_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S)),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_6262_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_6263_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_6264_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_6265_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_6266_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_6267_one__add__floor,axiom,
    ! [Xd: rat] : aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),one_one_int) = archim3151403230148437115or_rat(plus_plus_rat(Xd,one_one_rat)) ).

% one_add_floor
tff(fact_6268_one__add__floor,axiom,
    ! [Xd: real] : aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)) ).

% one_add_floor
tff(fact_6269_floor__log__eq__powr__iff,axiom,
    ! [Xd: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),Xd)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_6270_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Nb: nat] : archim3151403230148437115or_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_6271_floor__divide__of__nat__eq,axiom,
    ! [M: nat,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_6272_powr__realpow,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb) ) ) ).

% powr_realpow
tff(fact_6273_less__log__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd) ) ) ) ).

% less_log_iff
tff(fact_6274_log__less__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_6275_less__powr__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y) ) ) ) ).

% less_powr_iff
tff(fact_6276_powr__less__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ) ).

% powr_less_iff
tff(fact_6277_ceiling__altdef,axiom,
    ! [Xd: real] :
      aa_real_int(archim7802044766580827645g_real,Xd) = $ite(Xd = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xd)),aa_real_int(archim6058952711729229775r_real,Xd),aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int)) ).

% ceiling_altdef
tff(fact_6278_ceiling__diff__floor__le__1,axiom,
    ! [Xd: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xd)),aa_real_int(archim6058952711729229775r_real,Xd))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_6279_floor__eq,axiom,
    ! [Nb: int,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd) = Nb ) ) ) ).

% floor_eq
tff(fact_6280_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_6281_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_6282_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_6283_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_6284_zdvd__mult__cancel1,axiom,
    ! [M: int,Nb: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(M),Nb)),M)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6285_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_6286_floor__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,archim3151403230148437115or_rat(T))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I2)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I2),one_one_rat)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_6287_floor__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,T))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I2)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_6288_floor__eq__iff,axiom,
    ! [Xd: rat,A: int] :
      ( ( archim3151403230148437115or_rat(Xd) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xd)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_6289_floor__eq__iff,axiom,
    ! [Xd: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xd) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xd)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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_6290_floor__unique,axiom,
    ! [Z: int,Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xd)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat))
       => ( archim3151403230148437115or_rat(Xd) = Z ) ) ) ).

% floor_unique
tff(fact_6291_floor__unique,axiom,
    ! [Z: int,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xd) = Z ) ) ) ).

% floor_unique
tff(fact_6292_powr__minus__divide,axiom,
    ! [Xd: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),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,Xd),A)) ).

% powr_minus_divide
tff(fact_6293_less__floor__iff,axiom,
    ! [Z: int,Xd: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(Xd))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xd) ) ).

% less_floor_iff
tff(fact_6294_less__floor__iff,axiom,
    ! [Z: int,Xd: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xd) ) ).

% less_floor_iff
tff(fact_6295_floor__le__iff,axiom,
    ! [Xd: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xd)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% floor_le_iff
tff(fact_6296_floor__le__iff,axiom,
    ! [Xd: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xd)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_6297_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B))) ) ) ).

% le_mult_floor
tff(fact_6298_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% le_mult_floor
tff(fact_6299_binomial__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_6300_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),numeral_numeral_nat(bit0(one))))) ).

% binomial_maximum
tff(fact_6301_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,numeral_numeral_nat(bit0(one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_6302_binomial__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K4)),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),K4)) ) ) ).

% binomial_mono
tff(fact_6303_floor__correct,axiom,
    ! [Xd: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xd))),Xd)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),one_one_int))) ) ).

% floor_correct
tff(fact_6304_floor__correct,axiom,
    ! [Xd: 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,Xd))),Xd)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),one_one_int))) ) ).

% floor_correct
tff(fact_6305_powr__neg__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xd) ) ) ).

% powr_neg_one
tff(fact_6306_powr__mult__base,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_6307_le__log__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd) ) ) ) ).

% le_log_iff
tff(fact_6308_log__le__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_6309_le__powr__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y) ) ) ) ).

% le_powr_iff
tff(fact_6310_powr__le__iff,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ) ).

% powr_le_iff
tff(fact_6311_floor__eq2,axiom,
    ! [Nb: int,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd) = Nb ) ) ) ).

% floor_eq2
tff(fact_6312_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_6313_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,bit0(one))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_abs_add_iff
tff(fact_6314_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,bit0(one))),aa_int_int(plus_plus_int(K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_add_abs_iff
tff(fact_6315_floor__divide__lower,axiom,
    ! [Q2: rat,P3: 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(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P3,Q2)))),Q2)),P3) ) ).

% floor_divide_lower
tff(fact_6316_floor__divide__lower,axiom,
    ! [Q2: real,P3: 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,P3),Q2)))),Q2)),P3) ) ).

% floor_divide_lower
tff(fact_6317_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),numeral_numeral_nat(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_6318_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xd) = numeral_numeral_nat(bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_6319_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_6320_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K4)),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),K4)) ) ) ).

% binomial_strict_mono
tff(fact_6321_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))) ) ) ).

% central_binomial_odd
tff(fact_6322_ln__powr__bound,axiom,
    ! [Xd: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => ( 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,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_6323_ln__powr__bound2,axiom,
    ! [Xd: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => ( 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,Xd)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xd)) ) ) ).

% ln_powr_bound2
tff(fact_6324_log__add__eq__powr,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_6325_add__log__eq__powr,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_6326_minus__log__eq__powr,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xd)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_6327_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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(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,M),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),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,M),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_6328_incr__lemma,axiom,
    ! [D: int,Z: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(plus_plus_int(Xd),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xd),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_6329_decr__lemma,axiom,
    ! [D: int,Xd: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(Xd),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xd),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_6330_floor__divide__upper,axiom,
    ! [Q2: rat,P3: 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,P3),aa_rat_rat(times_times_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P3,Q2))),one_one_rat)),Q2)) ) ).

% floor_divide_upper
tff(fact_6331_floor__divide__upper,axiom,
    ! [Q2: real,P3: 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,P3),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,P3),Q2)))),one_one_real)),Q2)) ) ).

% floor_divide_upper
tff(fact_6332_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),Xd) = numeral_numeral_nat(bit0(one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_6333_round__def,axiom,
    ! [Xd: rat] : archim7778729529865785530nd_rat(Xd) = archim3151403230148437115or_rat(plus_plus_rat(Xd,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))) ).

% round_def
tff(fact_6334_round__def,axiom,
    ! [Xd: real] : aa_real_int(archim8280529875227126926d_real,Xd) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))) ).

% round_def
tff(fact_6335_log__minus__eq__powr,axiom,
    ! [B: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_6336_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(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_6337_powr__neg__numeral,axiom,
    ! [Xd: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(Nb))) ) ) ).

% powr_neg_numeral
tff(fact_6338_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(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_6339_arctan__add,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),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,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)))) ) ) ) ).

% arctan_add
tff(fact_6340_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_6341_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6342_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_6343_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))) ).

% choose_two
tff(fact_6344_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xd) ) ) ) ).

% both_member_options_ding
tff(fact_6345_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_6346_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_6347_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_6348_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_nat_o,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_o(P,Xd)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xd)),I)) ) ) ).

% intind
tff(fact_6349_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBTi_o,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBTi_o(P,Xd)
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(replicate_VEBT_VEBTi(Nb,Xd),I)) ) ) ).

% intind
tff(fact_6350_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_int_o,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_int_o(P,Xd)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xd)),I)) ) ) ).

% intind
tff(fact_6351_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBT_o(P,Xd)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xd),I)) ) ) ).

% intind
tff(fact_6352_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_6353_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_6354_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_6355_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_6356_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xd)),I) = Xd ) ) ).

% nth_replicate
tff(fact_6357_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBTi(replicate_VEBT_VEBTi(Nb,Xd),I) = Xd ) ) ).

% nth_replicate
tff(fact_6358_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xd)),I) = Xd ) ) ).

% nth_replicate
tff(fact_6359_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xd),I) = Xd ) ) ).

% nth_replicate
tff(fact_6360_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_6361_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_6362_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_6363_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_6364_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_6365_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_6366_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_6367_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_6368_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_6369_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,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,M),K))) ) ) ) ).

% choose_mult
tff(fact_6370_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_6371_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_6372_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_6373_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(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_6374_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,numeral_numeral_nat(bit0(one))),Nb)) ).

% binomial_le_pow2
tff(fact_6375_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_6376_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_6377_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_6378_in__children__def,axiom,
    ! [Nb: nat,TreeList: list_VEBT_VEBT,Xd: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeList),Xd)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),Nb)) ) ).

% in_children_def
tff(fact_6379_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xca: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xaa,Tree_is)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xba))),
                  pure_assn(( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                    & ( Nb = Nb ) )))),pure_assn(Xca = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xba)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeList),Summary)),Xca)),H) ) ).

% heaphelp
tff(fact_6380_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xca: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xaa,Tree_is)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xba))),
                  pure_assn(( ( none_nat = none_nat )
                    & ( Nb = Nb ) )))),pure_assn(Xca = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xba)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeList),Summary)),Xca)),H) ) ).

% heaphelp
tff(fact_6381_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xca: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xaa,Tree_is)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xba))),
                  pure_assn(( ( none_num = none_num )
                    & ( Nb = Nb ) )))),pure_assn(Xca = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xba)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeList),Summary)),Xca)),H) ) ).

% heaphelp
tff(fact_6382_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_6383_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_6384_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S) ).

% vebt_insert.simps(3)
tff(fact_6385_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary))) ) ).

% height_compose_child
tff(fact_6386_Rep__assn__inject,axiom,
    ! [Xd: assn,Y: assn] :
      ( ( rep_assn(Xd) = rep_assn(Y) )
    <=> ( Xd = Y ) ) ).

% Rep_assn_inject
tff(fact_6387_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_6388_List_Ofinite__set,axiom,
    ! [Xs: list_real] : aa_set_real_o(finite_finite_real,set_real2(Xs)) ).

% List.finite_set
tff(fact_6389_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_6390_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_6391_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_6392_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_Code_integer2(Xs)) ).

% List.finite_set
tff(fact_6393_in__set__replicate,axiom,
    ! [Xd: complex,Nb: nat,Y: complex] :
      ( aa_set_complex_o(member_complex(Xd),set_complex2(replicate_complex(Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6394_in__set__replicate,axiom,
    ! [Xd: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xd),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6395_in__set__replicate,axiom,
    ! [Xd: real,Nb: nat,Y: real] :
      ( aa_set_real_o(member_real(Xd),set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6396_in__set__replicate,axiom,
    ! [Xd: int,Nb: nat,Y: int] :
      ( aa_set_int_o(member_int(Xd),set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6397_in__set__replicate,axiom,
    ! [Xd: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xd = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6398_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X5) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6399_Bex__set__replicate,axiom,
    ! [Nb: nat,A: real,P: fun_real_o] :
      ( ? [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(replicate_real(Nb,A)))
          & aa_real_o(P,X5) )
    <=> ( aa_real_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6400_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X5) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6401_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X5) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6402_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X5) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6403_Ball__set__replicate,axiom,
    ! [Nb: nat,A: real,P: fun_real_o] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(replicate_real(Nb,A)))
         => aa_real_o(P,X5) )
    <=> ( aa_real_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6404_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X5) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6405_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X5) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6406_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),pure_assn((B)))),H)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_6407_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),pure_assn((B))))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ) ).

% ent_pure_post_iff
tff(fact_6408_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),pure_assn((B)))
    <=> ( ! [H2: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,P),one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_6409_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(A),B)),H)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B),H_2) ) ) ).

% mod_starE
tff(fact_6410_mod__starD,axiom,
    ! [A2: assn,B3: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(A2),B3)),H)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A2),H1)
          & aa_Pro653082453696054616_nat_o(rep_assn(B3),H22) ) ) ).

% mod_starD
tff(fact_6411_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
    <=> ! [H2: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H2)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q),H2) ) ) ).

% entails_def
tff(fact_6412_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H3)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q),H3) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ).

% entailsI
tff(fact_6413_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q),H) ) ) ).

% entailsD
tff(fact_6414_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q),H) ) ) ).

% ent_fwd
tff(fact_6415_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_6416_finite__list,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ? [Xs2: list_real] : set_real2(Xs2) = A2 ) ).

% finite_list
tff(fact_6417_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_6418_finite__list,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_6419_finite__list,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_6420_finite__list,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ? [Xs2: list_Code_integer] : set_Code_integer2(Xs2) = A2 ) ).

% finite_list
tff(fact_6421_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B3)
    <=> ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X5),B3) ) ) ).

% subset_code(1)
tff(fact_6422_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B3)
    <=> ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(Xs))
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),B3) ) ) ).

% subset_code(1)
tff(fact_6423_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B3)
    <=> ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(Xs))
         => aa_set_real_o(member_real(X5),B3) ) ) ).

% subset_code(1)
tff(fact_6424_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B3)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(Xs))
         => aa_set_int_o(member_int(X5),B3) ) ) ).

% subset_code(1)
tff(fact_6425_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B3)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X5),B3) ) ) ).

% subset_code(1)
tff(fact_6426_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_complex,Xsi2: list_complex,A2: fun_co6163879640491479806x_assn,A5: fun_co6163879640491479806x_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: complex,Xi: complex] :
              ( aa_set_complex_o(member_complex(X),set_complex2(Xs3))
             => ( aa_set_complex_o(member_complex(Xi),set_complex2(Xsi2))
               => ( aa_complex_assn(aa_com5932308198457167295x_assn(A2,X),Xi) = aa_complex_assn(aa_com5932308198457167295x_assn(A5,X),Xi) ) ) )
         => ( vEBT_L4260503343685368993omplex(A2,Xs,Xsi) = vEBT_L4260503343685368993omplex(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6427_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_co2031555705788908698T_assn,A5: fun_co2031555705788908698T_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: complex,Xi: vEBT_VEBT] :
              ( aa_set_complex_o(member_complex(X),set_complex2(Xs3))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A2,X),Xi) = aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A5,X),Xi) ) ) )
         => ( vEBT_L8524933119956041985T_VEBT(A2,Xs,Xsi) = vEBT_L8524933119956041985T_VEBT(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6428_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_nat,Xsi2: list_nat,A2: fun_co1748589094918083360t_assn,A5: fun_co1748589094918083360t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: complex,Xi: nat] :
              ( aa_set_complex_o(member_complex(X),set_complex2(Xs3))
             => ( aa_set_nat_o(member_nat(Xi),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_com1546947087700170465t_assn(A2,X),Xi) = aa_nat_assn(aa_com1546947087700170465t_assn(A5,X),Xi) ) ) )
         => ( vEBT_L137475477348087235ex_nat(A2,Xs,Xsi) = vEBT_L137475477348087235ex_nat(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6429_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_real,Xsi2: list_real,A2: fun_co8000059489268409724l_assn,A5: fun_co8000059489268409724l_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: complex,Xi: real] :
              ( aa_set_complex_o(member_complex(X),set_complex2(Xs3))
             => ( aa_set_real_o(member_real(Xi),set_real2(Xsi2))
               => ( aa_real_assn(aa_com7355656519465148733l_assn(A2,X),Xi) = aa_real_assn(aa_com7355656519465148733l_assn(A5,X),Xi) ) ) )
         => ( vEBT_L2479436891206192927x_real(A2,Xs,Xsi) = vEBT_L2479436891206192927x_real(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6430_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs3: list_complex,Xsi: list_int,Xsi2: list_int,A2: fun_co6164383388265235324t_assn,A5: fun_co6164383388265235324t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: complex,Xi: int] :
              ( aa_set_complex_o(member_complex(X),set_complex2(Xs3))
             => ( aa_set_int_o(member_int(Xi),set_int2(Xsi2))
               => ( aa_int_assn(aa_com2522482552906347837t_assn(A2,X),Xi) = aa_int_assn(aa_com2522482552906347837t_assn(A5,X),Xi) ) ) )
         => ( vEBT_L134985006839036959ex_int(A2,Xs,Xsi) = vEBT_L134985006839036959ex_int(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6431_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_complex,Xsi2: list_complex,A2: fun_VE496882281059548826x_assn,A5: fun_VE496882281059548826x_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: vEBT_VEBT,Xi: complex] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs3))
             => ( aa_set_complex_o(member_complex(Xi),set_complex2(Xsi2))
               => ( aa_complex_assn(aa_VEB2594926317323433291x_assn(A2,X),Xi) = aa_complex_assn(aa_VEB2594926317323433291x_assn(A5,X),Xi) ) ) )
         => ( vEBT_L2162147798726695391omplex(A2,Xs,Xsi) = vEBT_L2162147798726695391omplex(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6432_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: vEBT_VEBT,Xi: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs3))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A2,X),Xi) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A5,X),Xi) ) ) )
         => ( vEBT_L1279224858307276611T_VEBT(A2,Xs,Xsi) = vEBT_L1279224858307276611T_VEBT(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6433_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: vEBT_VEBT,Xi: nat] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs3))
             => ( aa_set_nat_o(member_nat(Xi),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A2,X),Xi) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A5,X),Xi) ) ) )
         => ( vEBT_L8296926524756676353BT_nat(A2,Xs,Xsi) = vEBT_L8296926524756676353BT_nat(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6434_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: vEBT_VEBT,Xi: real] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs3))
             => ( aa_set_real_o(member_real(Xi),set_real2(Xsi2))
               => ( aa_real_assn(aa_VEB8575627266564790985l_assn(A2,X),Xi) = aa_real_assn(aa_VEB8575627266564790985l_assn(A5,X),Xi) ) ) )
         => ( vEBT_L5781919052683127133T_real(A2,Xs,Xsi) = vEBT_L5781919052683127133T_real(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6435_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs3: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: fun_VE123837653852366616t_assn,A5: fun_VE123837653852366616t_assn] :
      ( ( Xs = Xs3 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X: vEBT_VEBT,Xi: int] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs3))
             => ( aa_set_int_o(member_int(Xi),set_int2(Xsi2))
               => ( aa_int_assn(aa_VEB392321844297299145t_assn(A2,X),Xi) = aa_int_assn(aa_VEB392321844297299145t_assn(A5,X),Xi) ) ) )
         => ( vEBT_L8294436054247626077BT_int(A2,Xs,Xsi) = vEBT_L8294436054247626077BT_int(A5,Xs3,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_6436_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S) ).

% vebt_insert.simps(2)
tff(fact_6437_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,Nb: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_6438_arcosh__def,axiom,
    ! [Xd: real] : aa_real_real(arcosh_real,Xd) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),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,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arcosh_def
tff(fact_6439_round__altdef,axiom,
    ! [Xd: rat] :
      archim7778729529865785530nd_rat(Xd) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))),archimedean_frac_rat(Xd)),archim2889992004027027881ng_rat(Xd),archim3151403230148437115or_rat(Xd)) ).

% round_altdef
tff(fact_6440_round__altdef,axiom,
    ! [Xd: real] :
      aa_real_int(archim8280529875227126926d_real,Xd) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_real_real(archim2898591450579166408c_real,Xd)),aa_real_int(archim7802044766580827645g_real,Xd),aa_real_int(archim6058952711729229775r_real,Xd)) ).

% round_altdef
tff(fact_6441_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xd: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xd) ).

% vebt_member.simps(2)
tff(fact_6442_arsinh__def,axiom,
    ! [Xd: real] : aa_real_real(arsinh_real,Xd) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),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,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))))) ).

% arsinh_def
tff(fact_6443_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_6444_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Nb: nat] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% inthall
tff(fact_6445_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),set_complex2(Xs))
         => aa_complex_o(P,X) )
     => ( 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_6446_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
     => ( 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_6447_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(P,X) )
     => ( 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_6448_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_o_o(P,(X)) )
     => ( 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_6449_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(P,X) )
     => ( 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_6450_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_int_o(P,X) )
     => ( 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_6451_length__replicate,axiom,
    ! [Nb: nat,Xd: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xd)) = Nb ).

% length_replicate
tff(fact_6452_length__replicate,axiom,
    ! [Nb: nat,Xd: real] : size_size_list_real(replicate_real(Nb,Xd)) = Nb ).

% length_replicate
tff(fact_6453_length__replicate,axiom,
    ! [Nb: nat,Xd: $o] : size_size_list_o(replicate_o(Nb,(Xd))) = Nb ).

% length_replicate
tff(fact_6454_length__replicate,axiom,
    ! [Nb: nat,Xd: nat] : size_size_list_nat(replicate_nat(Nb,Xd)) = Nb ).

% length_replicate
tff(fact_6455_length__replicate,axiom,
    ! [Nb: nat,Xd: int] : size_size_list_int(replicate_int(Nb,Xd)) = Nb ).

% length_replicate
tff(fact_6456_of__real__eq__0__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6457_of__real__eq__0__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xd) = zero_zero_complex )
    <=> ( Xd = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_6458_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_6459_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_6460_of__real__eq__1__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xd) = one_one_real )
    <=> ( Xd = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6461_of__real__eq__1__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xd) = one_one_complex )
    <=> ( Xd = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6462_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_6463_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_6464_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% of_real_numeral
tff(fact_6465_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_6466_of__real__mult,axiom,
    ! [Xd: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,Xd)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_mult
tff(fact_6467_of__real__mult,axiom,
    ! [Xd: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,Xd)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_mult
tff(fact_6468_of__real__divide,axiom,
    ! [Xd: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xd)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_6469_of__real__divide,axiom,
    ! [Xd: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xd)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_6470_of__real__add,axiom,
    ! [Xd: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xd)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_6471_of__real__add,axiom,
    ! [Xd: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xd)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_6472_of__real__power,axiom,
    ! [Xd: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xd)),Nb) ).

% of_real_power
tff(fact_6473_of__real__power,axiom,
    ! [Xd: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,Xd)),Nb) ).

% of_real_power
tff(fact_6474_frac__of__int,axiom,
    ! [Z: int] : archimedean_frac_rat(aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_6475_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_6476_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).

% of_real_neg_numeral
tff(fact_6477_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_6478_norm__of__real__add1,axiom,
    ! [Xd: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xd)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)) ).

% norm_of_real_add1
tff(fact_6479_norm__of__real__add1,axiom,
    ! [Xd: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xd)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)) ).

% norm_of_real_add1
tff(fact_6480_norm__of__real__addn,axiom,
    ! [Xd: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xd)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_6481_norm__of__real__addn,axiom,
    ! [Xd: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xd)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_6482_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_6483_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_6484_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_6485_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_6486_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_6487_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_6488_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_6489_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_6490_size__neq__size__imp__neq,axiom,
    ! [Xd: list_real,Y: list_real] :
      ( ( size_size_list_real(Xd) != size_size_list_real(Y) )
     => ( Xd != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6491_size__neq__size__imp__neq,axiom,
    ! [Xd: list_o,Y: list_o] :
      ( ( size_size_list_o(Xd) != size_size_list_o(Y) )
     => ( Xd != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6492_size__neq__size__imp__neq,axiom,
    ! [Xd: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xd) != size_size_list_nat(Y) )
     => ( Xd != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6493_size__neq__size__imp__neq,axiom,
    ! [Xd: list_int,Y: list_int] :
      ( ( size_size_list_int(Xd) != size_size_list_int(Y) )
     => ( Xd != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6494_size__neq__size__imp__neq,axiom,
    ! [Xd: num,Y: num] :
      ( ( size_size_num(Xd) != size_size_num(Y) )
     => ( Xd != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_6495_length__induct,axiom,
    ! [P: fun_list_real_o,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2))
             => aa_list_real_o(P,Ys2) )
         => aa_list_real_o(P,Xs2) )
     => aa_list_real_o(P,Xs) ) ).

% length_induct
tff(fact_6496_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_6497_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_6498_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_6499_finite__maxlen,axiom,
    ! [M7: set_list_real] :
      ( finite306553202115118035t_real(M7)
     => ? [N: nat] :
        ! [X4: list_real] :
          ( member_list_real(X4,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(X4)),N) ) ) ).

% finite_maxlen
tff(fact_6500_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( finite_finite_list_o(M7)
     => ? [N: nat] :
        ! [X4: list_o] :
          ( member_list_o(X4,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X4)),N) ) ) ).

% finite_maxlen
tff(fact_6501_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X4)),N) ) ) ).

% finite_maxlen
tff(fact_6502_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( finite3922522038869484883st_int(M7)
     => ? [N: nat] :
        ! [X4: list_int] :
          ( member_list_int(X4,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X4)),N) ) ) ).

% finite_maxlen
tff(fact_6503_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6504_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I3) = nth_VEBT_VEBTi(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6505_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I3) = aa_nat_real(nth_real(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6506_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6507_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6508_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_6509_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) )
    <=> ? [Xs4: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs4) = 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(Xs4,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6510_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na154763851189438853EBTi_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: vEBT_VEBTi] : aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),X_13) )
    <=> ? [Xs4: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs4) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),nth_VEBT_VEBTi(Xs4,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6511_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_real_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: real] : aa_real_o(aa_nat_fun_real_o(P,I2),X_13) )
    <=> ? [Xs4: list_real] :
          ( ( size_size_list_real(Xs4) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_real_o(aa_nat_fun_real_o(P,I2),aa_nat_real(nth_real(Xs4),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6512_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)) )
    <=> ? [Xs4: list_o] :
          ( ( size_size_list_o(Xs4) = 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(Xs4),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6513_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) )
    <=> ? [Xs4: list_nat] :
          ( ( size_size_list_nat(Xs4) = 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(Xs4),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6514_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) )
    <=> ? [Xs4: list_int] :
          ( ( size_size_list_int(Xs4) = 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(Xs4),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_6515_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6516_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6517_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [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) = aa_nat_real(nth_real(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6518_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6519_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6520_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6521_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6480751511521678896_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: vEBT_VEBT] : aa_nat_o(aa_VEB2314620471664397097_nat_o(P,Li),I3) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(P,nth_VEBT_VEBT(L3,I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6522_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6346650636051888901_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: vEBT_VEBTi] : aa_nat_o(aa_VEB8883426185042993134_nat_o(P,Li),I3) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_VEB8883426185042993134_nat_o(P,nth_VEBT_VEBTi(L3,I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6523_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_real_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: real] : aa_nat_o(aa_real_fun_nat_o(P,Li),I3) )
     => ~ ! [L3: list_real] :
            ( ( size_size_list_real(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_real_fun_nat_o(P,aa_nat_real(nth_real(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6524_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_o_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(P,(Li)),I3) )
     => ~ ! [L3: list_o] :
            ( ( size_size_list_o(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_o_fun_nat_o(P,aa_nat_o(nth_o(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6525_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(P,Li),I3) )
     => ~ ! [L3: list_nat] :
            ( ( size_size_list_nat(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(nth_nat(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6526_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_int_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(P,Li),I3) )
     => ~ ! [L3: list_int] :
            ( ( size_size_list_int(L3) = Nb )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Nb)
                 => aa_nat_o(aa_int_fun_nat_o(P,aa_nat_int(nth_int(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_6527_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xd: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y2: complex] :
            ( aa_set_complex_o(member_complex(Y2),set_complex2(Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate_complex(Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_6528_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xd: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y2),set_VEBT_VEBT2(Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_6529_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xd: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y2: real] :
            ( aa_set_real_o(member_real(Y2),set_real2(Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate_real(Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_6530_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xd: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y2: $o] :
            ( member_o((Y2),set_o2(Xs))
           => ( (Y2)
            <=> (Xd) ) )
       => ( Xs = replicate_o(Nb,(Xd)) ) ) ) ).

% replicate_eqI
tff(fact_6531_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xd: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate_nat(Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_6532_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xd: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),set_int2(Xs))
           => ( Y2 = Xd ) )
       => ( Xs = replicate_int(Nb,Xd) ) ) ) ).

% replicate_eqI
tff(fact_6533_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( X = Xd ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xd) = Xs ) ) ).

% replicate_length_same
tff(fact_6534_replicate__length__same,axiom,
    ! [Xs: list_real,Xd: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ( X = Xd ) )
     => ( replicate_real(size_size_list_real(Xs),Xd) = Xs ) ) ).

% replicate_length_same
tff(fact_6535_replicate__length__same,axiom,
    ! [Xs: list_o,Xd: $o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => ( (X)
          <=> (Xd) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xd)) = Xs ) ) ).

% replicate_length_same
tff(fact_6536_replicate__length__same,axiom,
    ! [Xs: list_nat,Xd: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( X = Xd ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xd) = Xs ) ) ).

% replicate_length_same
tff(fact_6537_replicate__length__same,axiom,
    ! [Xs: list_int,Xd: int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => ( X = Xd ) )
     => ( replicate_int(size_size_list_int(Xs),Xd) = Xs ) ) ).

% replicate_length_same
tff(fact_6538_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1930518968523514909l_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6539_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_real_assn,Xs: list_o,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4725278957065240257o_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6540_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6102073776069194049t_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6541_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_in6446958447386763770l_assn,Xs: list_int,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L8288995350762215837t_real(A2,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6542_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_real_fun_o_assn,Xs: list_real,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6234343332106409831real_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6543_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_o_assn,Xs: list_o,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7363604446928714179sn_o_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6544_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_nat_fun_o_assn,Xs: list_nat,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7887682484454631235_nat_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6545_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_int_fun_o_assn,Xs: list_int,Xsi: list_o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6066640139021943271_int_o(A2,Xs,Xsi)),H)
     => ( size_size_list_o(Xsi) = size_size_list_int(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6546_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1446010312343316929al_nat(A2,Xs,Xsi)),H)
     => ( size_size_list_nat(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6547_extract__pre__list__assn__lengthD,axiom,
    ! [A2: fun_o_fun_nat_assn,Xs: list_o,Xsi: list_nat,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4785011123346445925_o_nat(A2,Xs,Xsi)),H)
     => ( size_size_list_nat(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_6548_frac__ge__0,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xd)) ).

% frac_ge_0
tff(fact_6549_frac__ge__0,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),archimedean_frac_rat(Xd)) ).

% frac_ge_0
tff(fact_6550_frac__lt__1,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xd)),one_one_real) ).

% frac_lt_1
tff(fact_6551_frac__lt__1,axiom,
    ! [Xd: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,archimedean_frac_rat(Xd)),one_one_rat) ).

% frac_lt_1
tff(fact_6552_frac__1__eq,axiom,
    ! [Xd: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),one_one_real)) = aa_real_real(archim2898591450579166408c_real,Xd) ).

% frac_1_eq
tff(fact_6553_frac__1__eq,axiom,
    ! [Xd: rat] : archimedean_frac_rat(plus_plus_rat(Xd,one_one_rat)) = archimedean_frac_rat(Xd) ).

% frac_1_eq
tff(fact_6554_nonzero__of__real__divide,axiom,
    ! [Y: real,Xd: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xd)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_6555_nonzero__of__real__divide,axiom,
    ! [Y: real,Xd: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xd)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_6556_length__pos__if__in__set,axiom,
    ! [Xd: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xd),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_6557_length__pos__if__in__set,axiom,
    ! [Xd: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),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_6558_length__pos__if__in__set,axiom,
    ! [Xd: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xd),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_6559_length__pos__if__in__set,axiom,
    ! [Xd: $o,Xs: list_o] :
      ( member_o((Xd),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_6560_length__pos__if__in__set,axiom,
    ! [Xd: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xd),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_6561_length__pos__if__in__set,axiom,
    ! [Xd: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xd),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_6562_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_6563_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X5) )
    <=> ! [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_6564_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_o] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(Xs))
         => aa_real_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_6565_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X5: $o] :
          ( member_o((X5),set_o2(Xs))
         => aa_o_o(P,(X5)) )
    <=> ! [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_6566_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => aa_nat_o(P,X5) )
    <=> ! [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_6567_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(Xs))
         => aa_int_o(P,X5) )
    <=> ! [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_6568_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Xd: vEBT_VEBTi] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I3)) )
     => ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(Xs))
       => aa_VEBT_VEBTi_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6569_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xd: 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)) )
     => ( aa_set_complex_o(member_complex(Xd),set_complex2(Xs))
       => aa_complex_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6570_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xd: 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)) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6571_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xd: 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)) )
     => ( aa_set_real_o(member_real(Xd),set_real2(Xs))
       => aa_real_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6572_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xd: $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((Xd),set_o2(Xs))
       => aa_o_o(P,(Xd)) ) ) ).

% all_nth_imp_all_set
tff(fact_6573_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xd: 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)) )
     => ( aa_set_nat_o(member_nat(Xd),set_nat2(Xs))
       => aa_nat_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6574_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xd: 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)) )
     => ( aa_set_int_o(member_int(Xd),set_int2(Xs))
       => aa_int_o(P,Xd) ) ) ).

% all_nth_imp_all_set
tff(fact_6575_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(L))
         => aa_VEBT_VEBTi_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(L))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_6576_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(L))
         => aa_VEBT_VEBT_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(L))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_6577_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_o] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(L))
         => aa_real_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(L))
         => aa_real_o(P,aa_nat_real(nth_real(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_6578_all__set__conv__nth,axiom,
    ! [L: list_o,P: fun_o_o] :
      ( ! [X5: $o] :
          ( member_o((X5),set_o2(L))
         => aa_o_o(P,(X5)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(L))
         => aa_o_o(P,aa_nat_o(nth_o(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_6579_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(L))
         => aa_nat_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(L))
         => aa_nat_o(P,aa_nat_nat(nth_nat(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_6580_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_o] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),set_int2(L))
         => aa_int_o(P,X5) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(L))
         => aa_int_o(P,aa_nat_int(nth_int(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_6581_in__set__conv__nth,axiom,
    ! [Xd: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
          & ( nth_VEBT_VEBTi(Xs,I2) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6582_in__set__conv__nth,axiom,
    ! [Xd: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xd),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) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6583_in__set__conv__nth,axiom,
    ! [Xd: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),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) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6584_in__set__conv__nth,axiom,
    ! [Xd: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xd),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) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6585_in__set__conv__nth,axiom,
    ! [Xd: $o,Xs: list_o] :
      ( member_o((Xd),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)
          <=> (Xd) ) ) ) ).

% in_set_conv_nth
tff(fact_6586_in__set__conv__nth,axiom,
    ! [Xd: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xd),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) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6587_in__set__conv__nth,axiom,
    ! [Xd: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xd),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) = Xd ) ) ) ).

% in_set_conv_nth
tff(fact_6588_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( ! [X: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
           => aa_VEBT_VEBTi_o(P,X) )
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_6589_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))
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_6590_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_real,P: fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_real2(Xs))
           => aa_real_o(P,X) )
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_6591_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))
     => ( ! [X: $o] :
            ( member_o((X),set_o2(Xs))
           => aa_o_o(P,(X)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_6592_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))
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
           => aa_nat_o(P,X) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_6593_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))
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),set_int2(Xs))
           => aa_int_o(P,X) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_6594_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,Nb),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_6595_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_6596_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))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb)),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_6597_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_6598_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_6599_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_6600_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_6601_norm__less__p1,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xd)),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,Xd))),one_one_real))) ).

% norm_less_p1
tff(fact_6602_norm__less__p1,axiom,
    ! [Xd: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xd)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xd))),one_one_complex))) ).

% norm_less_p1
tff(fact_6603_frac__eq,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xd) = Xd )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ) ).

% frac_eq
tff(fact_6604_frac__eq,axiom,
    ! [Xd: rat] :
      ( ( archimedean_frac_rat(Xd) = Xd )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xd)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),one_one_rat) ) ) ).

% frac_eq
tff(fact_6605_frac__add,axiom,
    ! [Xd: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),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,Xd)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xd)),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,Xd)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ).

% frac_add
tff(fact_6606_frac__add,axiom,
    ! [Xd: rat,Y: rat] :
      archimedean_frac_rat(plus_plus_rat(Xd,Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(archimedean_frac_rat(Xd),archimedean_frac_rat(Y))),one_one_rat),plus_plus_rat(archimedean_frac_rat(Xd),archimedean_frac_rat(Y)),minus_minus_rat(plus_plus_rat(archimedean_frac_rat(Xd),archimedean_frac_rat(Y)),one_one_rat)) ).

% frac_add
tff(fact_6607_floor__add,axiom,
    ! [Xd: rat,Y: rat] :
      archim3151403230148437115or_rat(plus_plus_rat(Xd,Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(archimedean_frac_rat(Xd),archimedean_frac_rat(Y))),one_one_rat),aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),archim3151403230148437115or_rat(Y)),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(archim3151403230148437115or_rat(Xd)),archim3151403230148437115or_rat(Y))),one_one_int)) ).

% floor_add
tff(fact_6608_floor__add,axiom,
    ! [Xd: real,Y: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),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,Xd)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),aa_real_int(archim6058952711729229775r_real,Y)),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,Xd)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int)) ).

% floor_add
tff(fact_6609_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_6610_mod__frame__fwd,axiom,
    ! [Ps: assn,H: produc3658429121746597890et_nat,P: assn,R: assn,F2: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Ps),H)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,P),R)
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps),aa_assn_assn(times_times_assn(P),F2))
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(R),F2)),H) ) ) ) ).

% mod_frame_fwd
tff(fact_6611_log__base__10__eq1,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))))),aa_real_real(ln_ln_real,Xd)) ) ) ).

% log_base_10_eq1
tff(fact_6612_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_6613_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),one_one_nat)
    <=> ? [A6: $o,B6: $o] : T = vEBT_Leaf((A6),(B6)) ) ).

% deg1Leaf
tff(fact_6614_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),one_one_nat)
     => ? [A4: $o,B2: $o] : T = vEBT_Leaf((A4),(B2)) ) ).

% deg_1_Leaf
tff(fact_6615_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A4: $o,B2: $o] : T = vEBT_Leaf((A4),(B2)) ) ) ).

% deg_1_Leafy
tff(fact_6616_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_6617_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_6618_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_6619_exp__less__cancel__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% exp_less_cancel_iff
tff(fact_6620_exp__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_6621_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_6622_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_6623_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_6624_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_6625_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_6626_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_6627_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_6628_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_6629_exp__eq__one__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(exp_real,Xd) = one_one_real )
    <=> ( Xd = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_6630_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_6631_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_6632_exp__less__one__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xd)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_6633_one__less__exp__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% one_less_exp_iff
tff(fact_6634_exp__le__one__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xd)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_6635_one__le__exp__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% one_le_exp_iff
tff(fact_6636_exp__ln__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xd)) = Xd )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% exp_ln_iff
tff(fact_6637_exp__ln,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xd)) = Xd ) ) ).

% exp_ln
tff(fact_6638_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_6639_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_6640_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_6641_even__take__bit__eq,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),A) ) ) ).

% even_take_bit_eq
tff(fact_6642_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,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,bit0(one))),A) ) ) ).

% even_take_bit_eq
tff(fact_6643_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% take_bit_Suc_0
tff(fact_6644_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),numeral_numeral_nat(bit0(one))) ).

% take_bit_Suc_0
tff(fact_6645_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_Suc_0
tff(fact_6646_take__bit__of__exp,axiom,
    ! [M: nat,Nb: nat] : bit_se1745604003318907178nteger(M,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% take_bit_of_exp
tff(fact_6647_take__bit__of__exp,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% take_bit_of_exp
tff(fact_6648_take__bit__of__exp,axiom,
    ! [M: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(M),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% take_bit_of_exp
tff(fact_6649_take__bit__of__2,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(bit0(one))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Nb))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_of_2
tff(fact_6650_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Nb))),numeral_numeral_nat(bit0(one))) ).

% take_bit_of_2
tff(fact_6651_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_of_2
tff(fact_6652_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_6653_exp__less__cancel,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% exp_less_cancel
tff(fact_6654_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_6655_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(A),B)) ).

% take_bit_add
tff(fact_6656_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_6657_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(M),A) = aa_int_int(bit_se2923211474154528505it_int(M),B) ) ) ) ).

% take_bit_tightened
tff(fact_6658_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_6659_take__bit__of__nat,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)) ).

% take_bit_of_nat
tff(fact_6660_take__bit__of__nat,axiom,
    ! [Nb: nat,M: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)) ).

% take_bit_of_nat
tff(fact_6661_exp__not__eq__zero,axiom,
    ! [Xd: complex] : aa_complex_complex(exp_complex,Xd) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_6662_exp__not__eq__zero,axiom,
    ! [Xd: real] : aa_real_real(exp_real,Xd) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_6663_exp__times__arg__commute,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(exp_real,A2)) ).

% exp_times_arg_commute
tff(fact_6664_exp__times__arg__commute,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(exp_complex,A2)) ).

% exp_times_arg_commute
tff(fact_6665_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_6666_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_6667_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_6668_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] : aa_real_real(exp_real,X) = Y ) ).

% exp_total
tff(fact_6669_exp__gt__zero,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xd)) ).

% exp_gt_zero
tff(fact_6670_not__exp__less__zero,axiom,
    ! [Xd: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xd)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_6671_not__exp__le__zero,axiom,
    ! [Xd: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xd)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_6672_exp__ge__zero,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xd)) ).

% exp_ge_zero
tff(fact_6673_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6674_signed__take__bit__take__bit,axiom,
    ! [M: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(M),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),M),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(M)),A) ).

% signed_take_bit_take_bit
tff(fact_6675_exp__add__commuting,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xd) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_6676_exp__add__commuting,axiom,
    ! [Xd: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(Xd),Y) = aa_complex_complex(times_times_complex(Y),Xd) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xd),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_6677_mult__exp__exp,axiom,
    ! [Xd: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) ).

% mult_exp_exp
tff(fact_6678_mult__exp__exp,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xd),Y)) ).

% mult_exp_exp
tff(fact_6679_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_6680_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,M: 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,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_6681_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,M: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Nb,bit_se2793503036327961859nteger(M,A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),bit_se1745604003318907178nteger(Nb,A),bit_se2793503036327961859nteger(M,bit_se1745604003318907178nteger(Nb,A))) ).

% take_bit_set_bit_eq
tff(fact_6682_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,M: 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,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_6683_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_6684_exp__diff,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(Xd),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_6685_exp__diff,axiom,
    ! [Xd: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_6686_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,M: 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,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_6687_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,M: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(M),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_6688_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_6689_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_Leaf((A),(B))) = one_one_real ).

% VEBT_internal.cnt.simps(1)
tff(fact_6690_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_Leaf((A),(B))) = one_one_nat ).

% VEBT_internal.cnt'.simps(1)
tff(fact_6691_take__bit__signed__take__bit,axiom,
    ! [M: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_6692_exp__gt__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xd)) ) ).

% exp_gt_one
tff(fact_6693_exp__ge__add__one__self,axiom,
    ! [Xd: 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),Xd)),aa_real_real(exp_real,Xd)) ).

% exp_ge_add_one_self
tff(fact_6694_exp__minus__inverse,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xd))) = one_one_real ).

% exp_minus_inverse
tff(fact_6695_exp__minus__inverse,axiom,
    ! [Xd: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xd))) = one_one_complex ).

% exp_minus_inverse
tff(fact_6696_exp__of__nat2__mult,axiom,
    ! [Xd: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xd)),Nb) ).

% exp_of_nat2_mult
tff(fact_6697_exp__of__nat2__mult,axiom,
    ! [Xd: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(Xd),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xd)),Nb) ).

% exp_of_nat2_mult
tff(fact_6698_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xd: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xd)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xd)),Nb) ).

% exp_of_nat_mult
tff(fact_6699_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xd: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xd)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xd)),Nb) ).

% exp_of_nat_mult
tff(fact_6700_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_6701_log__ln,axiom,
    ! [Xd: real] : aa_real_real(ln_ln_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),Xd) ).

% log_ln
tff(fact_6702_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Leaf((A),(B))),Xd)
    <=> $ite(
          Xd = zero_zero_nat,
          (A),
          $ite(Xd = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_6703_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_6704_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xd) = $ite(
        Xd = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xd = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_6705_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_6706_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_6707_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_6708_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_6709_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Leaf((A),(B))),Xd)
    <=> $ite(
          Xd = zero_zero_nat,
          (A),
          $ite(Xd = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_6710_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_6711_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((Uu),(Uv))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_6712_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((Uu),(Uv))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_6713_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_6714_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_6715_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] : aa_nat_nat(vEBT_T_s_u_c_c2(vEBT_Leaf((Uu),(B))),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_6716_num_Osize_I4_J,axiom,
    size_size_num(one) = zero_zero_nat ).

% num.size(4)
tff(fact_6717_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C) = aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) ).

% star_assoc
tff(fact_6718_star__aci_I2_J,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(times_times_assn(A),B) = aa_assn_assn(times_times_assn(B),A) ).

% star_aci(2)
tff(fact_6719_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(A),aa_assn_assn(times_times_assn(B),C)) = aa_assn_assn(times_times_assn(B),aa_assn_assn(times_times_assn(A),C)) ).

% star_aci(3)
tff(fact_6720_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),C)),B) ).

% assn_aci(10)
tff(fact_6721_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ).

% is_entails
tff(fact_6722_exp__ge__add__one__self__aux,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xd)),aa_real_real(exp_real,Xd)) ) ).

% exp_ge_add_one_self_aux
tff(fact_6723_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_6724_ln__ge__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xd))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xd) ) ) ).

% ln_ge_iff
tff(fact_6725_ln__x__over__x__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),Xd)) ) ) ).

% ln_x_over_x_mono
tff(fact_6726_powr__def,axiom,
    ! [Xd: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),A) = $ite(Xd = 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,Xd)))) ).

% powr_def
tff(fact_6727_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_6728_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_6729_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf((A),(B))) = numeral_numeral_nat(bit0(bit0(one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_6730_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_6731_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_6732_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A),(B))),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(Xd = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_6733_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] : aa_nat_nat(vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_6734_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf((A),(B))) = numeral_numeral_nat(aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_6735_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),numera6620942414471956472nteger(bit0(K))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_6736_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)),numeral_numeral_nat(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),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_6737_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,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_6738_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(Nb,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% take_bit_eq_mod
tff(fact_6739_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,numeral_numeral_nat(bit0(one))),Nb)) ).

% take_bit_eq_mod
tff(fact_6740_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% take_bit_eq_mod
tff(fact_6741_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one))) ).

% exp_le
tff(fact_6742_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xd: 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,Xd),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xd) ) ) ).

% exp_divide_power_eq
tff(fact_6743_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xd: 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(Xd),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xd) ) ) ).

% exp_divide_power_eq
tff(fact_6744_tanh__altdef,axiom,
    ! [Xd: real] : aa_real_real(tanh_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xd)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xd)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xd)))) ).

% tanh_altdef
tff(fact_6745_tanh__altdef,axiom,
    ! [Xd: complex] : aa_complex_complex(tanh_complex,Xd) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xd)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xd)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xd)))) ).

% tanh_altdef
tff(fact_6746_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_6747_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Nb,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6748_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,numeral_numeral_nat(bit0(one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6749_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,bit0(one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6750_bin__last__bintrunc,axiom,
    ! [L: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,L),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),L)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ) ) ).

% bin_last_bintrunc
tff(fact_6751_bin__last__bintrunc,axiom,
    ! [L: nat,Nb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se2923211474154528505it_int(L),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),L)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Nb) ) ) ).

% bin_last_bintrunc
tff(fact_6752_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(L),numera6620942414471956472nteger(bit0(K))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(pred_numeral(L),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_6753_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_6754_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,bit0(K))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_6755_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),numeral_numeral_real(bit0(one))) ).

% exp_half_le2
tff(fact_6756_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),numeral_numeral_nat(bit0(one))) ).

% exp_double
tff(fact_6757_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),numeral_numeral_nat(bit0(one))) ).

% exp_double
tff(fact_6758_num_Osize_I5_J,axiom,
    ! [X2: num] : size_size_num(bit0(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_num(X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_6759_num_Osize_I6_J,axiom,
    ! [X32: num] : size_size_num(aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_6760_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),numeral_numeral_real(bit0(one))) ) ).

% exp_bound_half
tff(fact_6761_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),numeral_numeral_real(bit0(one))) ) ).

% exp_bound_half
tff(fact_6762_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A),(B))),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
        $ite(Xd = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_6763_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),numera6620942414471956472nteger(aa_num_num(bit1,K))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_6764_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)),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),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_6765_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_6766_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),uminus1351360451143612070nteger(one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),aa_nat_nat(suc,Nb)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_6767_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_6768_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(K),uminus1351360451143612070nteger(one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),numeral_numeral_nat(K)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_6769_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),numeral_numeral_nat(K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_6770_take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))),numera6620942414471956472nteger(bit0(one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% take_bit_Suc
tff(fact_6771_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),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one)))) ).

% take_bit_Suc
tff(fact_6772_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% take_bit_Suc
tff(fact_6773_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F2: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),R)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Ps),aa_assn_assn(times_times_assn(P),F2))
       => ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(R),F2)),Q)
         => aa_assn_o(aa_assn_fun_assn_o(entails,Ps),Q) ) ) ) ).

% ent_frame_fwd
tff(fact_6774_fr__rot__rhs,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(times_times_assn(B3),C5))
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(times_times_assn(C5),B3)) ) ).

% fr_rot_rhs
tff(fact_6775_fr__refl,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B3)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(A2),C5)),aa_assn_assn(times_times_assn(B3),C5)) ) ).

% fr_refl
tff(fact_6776_fr__rot,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(A2),B3)),C5)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(B3),A2)),C5) ) ).

% fr_rot
tff(fact_6777_exp__bound,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xd)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one))))) ) ) ).

% exp_bound
tff(fact_6778_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_6779_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(times_times_assn(A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_6780_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))) = A )
     => ( bit_se1745604003318907178nteger(Nb,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6781_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(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,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6782_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,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,bit0(one))),A),zero_zero_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6783_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(L),numera6620942414471956472nteger(aa_num_num(bit1,K))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(pred_numeral(L),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_6784_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),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,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_6785_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_6786_real__exp__bound__lemma,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd))) ) ) ).

% real_exp_bound_lemma
tff(fact_6787_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xd: 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))),Xd)
     => ( 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,Xd),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xd)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_6788_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xd))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_6789_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_6790_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_6791_exp__lower__Taylor__quadratic,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => 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),Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),numeral_numeral_real(bit0(one))))),aa_real_real(exp_real,Xd)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_6792_log__base__10__eq2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(aa_num_num(bit1,bit0(one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xd)) ) ) ).

% log_base_10_eq2
tff(fact_6793_tanh__real__altdef,axiom,
    ! [Xd: real] : aa_real_real(tanh_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),Xd)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),Xd)))) ).

% tanh_real_altdef
tff(fact_6794_take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Nb,A) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))),numera6620942414471956472nteger(bit0(one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% take_bit_rec
tff(fact_6795_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),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))))) ).

% take_bit_rec
tff(fact_6796_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,bit0(one))))),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))))) ).

% take_bit_rec
tff(fact_6797_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((A),(B))) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_6798_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6799_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6800_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_6801_uint32_Osize__eq,axiom,
    ! [P3: uint32] : size_size_uint32(P3) = numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))) ).

% uint32.size_eq
tff(fact_6802_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( nth_VEBT_VEBT(slice_VEBT_VEBT(From,To,Xs),I) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6803_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBTi,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( nth_VEBT_VEBTi(slice_VEBT_VEBTi(From,To,Xs),I) = nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6804_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_real,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_real(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_real(nth_real(slice_real(From,To,Xs)),I) = aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6805_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_o(nth_o(slice_o(From,To,Xs)),I)
          <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6806_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_nat(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6807_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6808_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = numeral_numeral_nat(K) ).

% pred_numeral_inc
tff(fact_6809_slice__complete,axiom,
    ! [Xs: list_real] : slice_real(zero_zero_nat,size_size_list_real(Xs),Xs) = Xs ).

% slice_complete
tff(fact_6810_slice__complete,axiom,
    ! [Xs: list_o] : slice_o(zero_zero_nat,size_size_list_o(Xs),Xs) = Xs ).

% slice_complete
tff(fact_6811_slice__complete,axiom,
    ! [Xs: list_nat] : slice_nat(zero_zero_nat,size_size_list_nat(Xs),Xs) = Xs ).

% slice_complete
tff(fact_6812_slice__complete,axiom,
    ! [Xs: list_int] : slice_int(zero_zero_nat,size_size_list_int(Xs),Xs) = Xs ).

% slice_complete
tff(fact_6813_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6814_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,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6815_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6816_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(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6817_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_plus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(numeral_numeral_rat(Nb))) = uminus_uminus_rat(numeral_numeral_rat(inc(Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6818_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_6819_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_6820_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_6821_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_6822_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(inc(M))) ).

% add_neg_numeral_special(6)
tff(fact_6823_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(Nb))) ).

% diff_numeral_special(5)
tff(fact_6824_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)),numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(Nb))) ).

% diff_numeral_special(5)
tff(fact_6825_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(inc(Nb))) ).

% diff_numeral_special(5)
tff(fact_6826_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(inc(Nb))) ).

% diff_numeral_special(5)
tff(fact_6827_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_minus_rat(uminus_uminus_rat(one_one_rat),numeral_numeral_rat(Nb)) = uminus_uminus_rat(numeral_numeral_rat(inc(Nb))) ).

% diff_numeral_special(5)
tff(fact_6828_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,inc(M)) ).

% diff_numeral_special(6)
tff(fact_6829_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(inc(M)) ).

% diff_numeral_special(6)
tff(fact_6830_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),uminus1351360451143612070nteger(one_one_Code_integer)) = numera6620942414471956472nteger(inc(M)) ).

% diff_numeral_special(6)
tff(fact_6831_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(inc(M)) ).

% diff_numeral_special(6)
tff(fact_6832_diff__numeral__special_I6_J,axiom,
    ! [M: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(inc(M)) ).

% diff_numeral_special(6)
tff(fact_6833_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_6834_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_6835_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,M: 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),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_6836_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_6837_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% take_bit_mult
tff(fact_6838_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% take_bit_diff
tff(fact_6839_num__induct,axiom,
    ! [P: fun_num_o,Xd: num] :
      ( aa_num_o(P,one)
     => ( ! [X: num] :
            ( aa_num_o(P,X)
           => aa_num_o(P,inc(X)) )
       => aa_num_o(P,Xd) ) ) ).

% num_induct
tff(fact_6840_add__inc,axiom,
    ! [Xd: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xd),inc(Y)) = inc(aa_num_num(aa_num_fun_num_num(plus_plus_num,Xd),Y)) ).

% add_inc
tff(fact_6841_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_6842_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_6843_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_6844_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_6845_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_6846_inc_Osimps_I1_J,axiom,
    inc(one) = bit0(one) ).

% inc.simps(1)
tff(fact_6847_inc_Osimps_I3_J,axiom,
    ! [Xd: num] : inc(aa_num_num(bit1,Xd)) = bit0(inc(Xd)) ).

% inc.simps(3)
tff(fact_6848_inc_Osimps_I2_J,axiom,
    ! [Xd: num] : inc(bit0(Xd)) = aa_num_num(bit1,Xd) ).

% inc.simps(2)
tff(fact_6849_add__One,axiom,
    ! [Xd: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xd),one) = inc(Xd) ).

% add_One
tff(fact_6850_mult__inc,axiom,
    ! [Xd: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xd),inc(Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xd),Y)),Xd) ).

% mult_inc
tff(fact_6851_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),one_one_int)) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_6852_numeral__inc,axiom,
    ! [Xd: num] : numera6690914467698888265omplex(inc(Xd)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xd)),one_one_complex) ).

% numeral_inc
tff(fact_6853_numeral__inc,axiom,
    ! [Xd: num] : numeral_numeral_real(inc(Xd)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Xd)),one_one_real) ).

% numeral_inc
tff(fact_6854_numeral__inc,axiom,
    ! [Xd: num] : numeral_numeral_rat(inc(Xd)) = plus_plus_rat(numeral_numeral_rat(Xd),one_one_rat) ).

% numeral_inc
tff(fact_6855_numeral__inc,axiom,
    ! [Xd: num] : numeral_numeral_nat(inc(Xd)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Xd)),one_one_nat) ).

% numeral_inc
tff(fact_6856_numeral__inc,axiom,
    ! [Xd: num] : aa_num_int(numeral_numeral_int,inc(Xd)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Xd)),one_one_int) ).

% numeral_inc
tff(fact_6857_take__bit__nat__eq__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_6858_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_6859_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_6860_take__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% take_bit_nat_def
tff(fact_6861_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_6862_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,bit0(one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_6863_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% take_bit_int_def
tff(fact_6864_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,bit0(one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_6865_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_6866_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,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_6867_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,bit0(one))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_6868_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,bit0(one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_6869_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,bit0(one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_6870_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,bit0(one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_6871_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_6872_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_numeral_minus_bit0
tff(fact_6873_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,bit0(one))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_6874_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_6875_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_6876_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,bit0(one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_6877_powr__int,axiom,
    ! [Xd: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),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,Xd),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,Xd),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_6878_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_6879_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_6880_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_6881_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_6882_set__decode__inverse,axiom,
    ! [Nb: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,Nb)) = Nb ).

% set_decode_inverse
tff(fact_6883_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_6884_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_6885_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_6886_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_6887_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_6888_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_nat(K) ).

% nat_numeral
tff(fact_6889_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_6890_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_6891_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_6892_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_6893_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_6894_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_6895_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_6896_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_6897_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_6898_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_6899_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_6900_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% fact_Suc
tff(fact_6901_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_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_6902_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_6903_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_6904_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_6905_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_6906_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_6907_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_6908_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_6909_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_6910_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_6911_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_6912_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_6913_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_6914_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_6915_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,numeral_numeral_nat(bit0(one))) = numeral_numeral_rat(bit0(one)) ).

% fact_2
tff(fact_6916_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,numeral_numeral_nat(bit0(one))) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% fact_2
tff(fact_6917_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,numeral_numeral_nat(bit0(one))) = numeral_numeral_nat(bit0(one)) ).

% fact_2
tff(fact_6918_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,numeral_numeral_nat(bit0(one))) = numeral_numeral_real(bit0(one)) ).

% fact_2
tff(fact_6919_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,numeral_numeral_nat(bit0(one))) = numera6690914467698888265omplex(bit0(one)) ).

% fact_2
tff(fact_6920_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_6921_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z)) = aa_int_real(ring_1_of_int_real,Z) ) ) ).

% of_nat_nat
tff(fact_6922_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = aa_int_int(ring_1_of_int_int,Z) ) ) ).

% of_nat_nat
tff(fact_6923_of__nat__nat,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,Z)) = aa_int_complex(ring_17405671764205052669omplex,Z) ) ) ).

% of_nat_nat
tff(fact_6924_diff__nat__numeral,axiom,
    ! [V: num,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),numeral_numeral_nat(V2)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V2))) ).

% diff_nat_numeral
tff(fact_6925_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xd: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_6926_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xd: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(Xd)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xd)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_6927_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ).

% dvd_nat_abs_iff
tff(fact_6928_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_6929_nat__ceiling__le__eq,axiom,
    ! [Xd: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xd))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_6930_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_6931_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_6932_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xd: 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,numeral_numeral_nat(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_6933_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,numeral_numeral_nat(Xd)),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,Xd)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_6934_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xd: 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,numeral_numeral_nat(Xd)),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,Xd)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_6935_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xd: 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,numeral_numeral_nat(Xd)),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,Xd)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_6936_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) != zero_zero_rat ).

% fact_nonzero
tff(fact_6937_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) != zero_zero_int ).

% fact_nonzero
tff(fact_6938_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) != zero_zero_nat ).

% fact_nonzero
tff(fact_6939_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) != zero_zero_real ).

% fact_nonzero
tff(fact_6940_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) != zero_zero_complex ).

% fact_nonzero
tff(fact_6941_fact__less__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_6942_nat__numeral__as__int,axiom,
    ! [X4: num] : numeral_numeral_nat(X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_6943_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_6944_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_6945_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_6946_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_6947_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_6948_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_6949_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_6950_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_6951_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_6952_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_6953_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_6954_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_6955_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_6956_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_6957_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_6958_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_6959_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_6960_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_6961_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
          & aa_nat_o(P,aa_int_nat(nat2,X5)) ) ) ).

% ex_nat
tff(fact_6962_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P,X_13)
    <=> ! [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => aa_nat_o(P,aa_int_nat(nat2,X5)) ) ) ).

% all_nat
tff(fact_6963_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_6964_fact__dvd,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,M)) ) ).

% fact_dvd
tff(fact_6965_fact__dvd,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) ) ).

% fact_dvd
tff(fact_6966_fact__dvd,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,M)) ) ).

% fact_dvd
tff(fact_6967_set__encode__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B3) )
        <=> ( A2 = B3 ) ) ) ) ).

% set_encode_eq
tff(fact_6968_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_6969_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_6970_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_6971_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_6972_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_6973_unset__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_6974_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_6975_dvd__fact,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_6976_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_6977_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_6978_fact__less__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_6979_fact__less__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_6980_fact__less__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_6981_fact__less__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_6982_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_6983_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_6984_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_6985_int__minus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_6986_fact__mod,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_6987_fact__mod,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_6988_fact__mod,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_6989_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_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_6990_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6991_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_6992_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_6993_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_6994_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_6995_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_6996_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_6997_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(times_times_int(W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_6998_real__nat__ceiling__ge,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xd)))) ).

% real_nat_ceiling_ge
tff(fact_6999_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_7000_fact__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),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,M)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_7001_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_7002_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,archim3151403230148437115or_rat(R2)))),R2) ) ).

% of_nat_floor
tff(fact_7003_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_7004_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_7005_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_7006_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_7007_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_7008_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N6: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N6) )
           => aa_nat_o(P,N6) )
        & ( 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_7009_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_7010_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( aa_int_nat(nat2,aa_int_int(plus_plus_int(Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_7011_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_7012_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).

% Suc_as_int
tff(fact_7013_le__mult__nat__floor,axiom,
    ! [A: real,B: 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,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ).

% le_mult_nat_floor
tff(fact_7014_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_7015_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z4),Z)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_7016_nat__diff__distrib_H,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Xd),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xd)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_7017_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_7018_nat__div__distrib_H,axiom,
    ! [Y: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(Xd),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xd)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_7019_nat__div__distrib,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(Xd),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xd)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_7020_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_7021_nat__mod__distrib,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(Xd),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xd)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_7022_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(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_7023_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(times_3573771949741848930nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_7024_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_7025_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_7026_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_7027_nat__floor__neg,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xd)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7028_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_7029_fact__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(semiri3624122377584611663nteger,numeral_numeral_nat(K)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(K)),aa_nat_Code_integer(semiri3624122377584611663nteger,pred_numeral(K))) ).

% fact_numeral
tff(fact_7030_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,numeral_numeral_nat(K)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,pred_numeral(K))) ).

% fact_numeral
tff(fact_7031_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,numeral_numeral_nat(K)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,pred_numeral(K))) ).

% fact_numeral
tff(fact_7032_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,numeral_numeral_nat(K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),aa_nat_nat(semiri1408675320244567234ct_nat,pred_numeral(K))) ).

% fact_numeral
tff(fact_7033_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,numeral_numeral_nat(K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K)),aa_nat_real(semiri2265585572941072030t_real,pred_numeral(K))) ).

% fact_numeral
tff(fact_7034_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,numeral_numeral_nat(K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,pred_numeral(K))) ).

% fact_numeral
tff(fact_7035_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_7036_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_7037_floor__eq3,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xd)) = Nb ) ) ) ).

% floor_eq3
tff(fact_7038_le__nat__floor,axiom,
    ! [Xd: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_7039_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_7040_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_7041_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_7042_nat__less__iff,axiom,
    ! [W: int,M: 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)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_7043_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7044_floor__eq4,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xd)) = Nb ) ) ) ).

% floor_eq4
tff(fact_7045_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,numeral_numeral_nat(bit0(one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_7046_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),uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7047_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),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_7048_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_7049_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_7050_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_7051_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_7052_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7053_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7054_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7055_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7056_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_real(semiri2265585572941072030t_real,M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7057_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7058_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7059_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(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_7060_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_7061_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_7062_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_7063_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_7064_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_7065_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(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_7066_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_7067_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_7068_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = 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_7069_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_7070_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_7071_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)) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_7072_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_7073_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_7074_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,numeral_numeral_nat(bit0(one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K) ) ) ).

% even_nat_iff
tff(fact_7075_sin__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(sin_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X4))) ).

% sin_coeff_def
tff(fact_7076_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,numeral_numeral_nat(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_7077_cos__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(cos_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X4)),zero_zero_real) ).

% cos_coeff_def
tff(fact_7078_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,numeral_numeral_nat(bit0(one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7079_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,numeral_numeral_nat(bit0(one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7080_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,numeral_numeral_nat(bit0(one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7081_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,numeral_numeral_nat(bit0(one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7082_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(divide_divide_int(K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_7083_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_7084_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_7085_sgn__sgn,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(sgn_sgn_Code_integer(A)) = sgn_sgn_Code_integer(A) ).

% sgn_sgn
tff(fact_7086_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_7087_sgn__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(sgn_sgn_rat(A)) = sgn_sgn_rat(A) ).

% sgn_sgn
tff(fact_7088_sgn__zero,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_zero
tff(fact_7089_sgn__zero,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_zero
tff(fact_7090_sgn__0,axiom,
    sgn_sgn_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_7091_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_7092_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_7093_sgn__0,axiom,
    sgn_sgn_rat(zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_7094_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_7095_sgn__one,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_one
tff(fact_7096_sgn__one,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_one
tff(fact_7097_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_7098_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_7099_sgn__1,axiom,
    sgn_sgn_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_7100_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_7101_sgn__1,axiom,
    sgn_sgn_rat(one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_7102_sgn__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_divide
tff(fact_7103_sgn__divide,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_divide
tff(fact_7104_sgn__divide,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(divide_divide_rat(A,B)) = divide_divide_rat(sgn_sgn_rat(A),sgn_sgn_rat(B)) ).

% sgn_divide
tff(fact_7105_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_7106_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_7107_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(uminus1351360451143612070nteger(A)) = uminus1351360451143612070nteger(sgn_sgn_Code_integer(A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7108_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_7109_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : sgn_sgn_rat(uminus_uminus_rat(A)) = uminus_uminus_rat(sgn_sgn_rat(A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_7110_power__sgn,axiom,
    ! [A: rat,Nb: nat] : sgn_sgn_rat(aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(sgn_sgn_rat(A)),Nb) ).

% power_sgn
tff(fact_7111_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_7112_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_7113_power__sgn,axiom,
    ! [A: code_integer,Nb: nat] : sgn_sgn_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(sgn_sgn_Code_integer(A)),Nb) ).

% power_sgn
tff(fact_7114_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,sgn_sgn_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_7115_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_7116_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_7117_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_7118_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),sgn_sgn_Code_integer(A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_7119_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_7120_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),sgn_sgn_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_7121_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_7122_divide__sgn,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(sgn_sgn_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(sgn_sgn_real,B)) ).

% divide_sgn
tff(fact_7123_divide__sgn,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,sgn_sgn_rat(B)) = aa_rat_rat(times_times_rat(A),sgn_sgn_rat(B)) ).

% divide_sgn
tff(fact_7124_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7125_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7126_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( sgn_sgn_Code_integer(A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_7127_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_7128_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( sgn_sgn_rat(A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_7129_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_7130_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_7131_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_7132_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(sgn_sgn_rat(A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_7133_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_7134_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_7135_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)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_7136_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_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_7137_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(sgn_sgn_Code_integer(A)),sgn_sgn_Code_integer(A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_7138_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_7139_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% sgn_abs
tff(fact_7140_sgn__abs,axiom,
    ! [A: rat] : abs_abs_rat(sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_abs
tff(fact_7141_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_7142_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_abs
tff(fact_7143_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_7144_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = aa_o_real(zero_n3304061248610475627l_real,A != zero_zero_real) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7145_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : sgn_sgn_rat(abs_abs_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7146_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_7147_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : sgn_sgn_Code_integer(abs_abs_Code_integer(A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_7148_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(times_times_int(K),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_7149_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_7150_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(L),aa_int_int(sgn_sgn_int,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_7151_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_7152_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_7153_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_7154_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( sgn_sgn_Code_integer(A) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_7155_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ).

% sgn_neg
tff(fact_7156_sgn__of__nat,axiom,
    ! [Nb: nat] : 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_7157_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_o_real(zero_n3304061248610475627l_real,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_7158_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_7159_sgn__of__nat,axiom,
    ! [Nb: nat] : sgn_sgn_Code_integer(aa_nat_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_7160_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_7161_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_7162_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_7163_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_7164_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_7165_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_7166_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_7167_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_7168_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_7169_sgn__zero__iff,axiom,
    ! [Xd: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,Xd) = zero_zero_complex )
    <=> ( Xd = zero_zero_complex ) ) ).

% sgn_zero_iff
tff(fact_7170_sgn__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sgn_sgn_real,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% sgn_zero_iff
tff(fact_7171_sgn__mult,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),sgn_sgn_rat(B)) ).

% sgn_mult
tff(fact_7172_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_7173_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A),B)) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_7174_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : sgn_sgn_Code_integer(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(A),B)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(sgn_sgn_Code_integer(A)),sgn_sgn_Code_integer(B)) ).

% sgn_mult
tff(fact_7175_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_7176_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [Xd: real,Y: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xd)),aa_real_real(sgn_sgn_real,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_7177_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(Xd),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,Xd)),aa_complex_complex(sgn_sgn_complex,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_7178_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B) = sgn_sgn_Code_integer(A) )
     => ( sgn_sgn_Code_integer(plus_p5714425477246183910nteger(A,B)) = sgn_sgn_Code_integer(A) ) ) ).

% same_sgn_sgn_add
tff(fact_7179_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7180_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( sgn_sgn_rat(plus_plus_rat(A,B)) = sgn_sgn_rat(A) ) ) ).

% same_sgn_sgn_add
tff(fact_7181_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_7182_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7183_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7184_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B) != sgn_sgn_Code_integer(A) )
     => ( ( sgn_sgn_Code_integer(A) != zero_z3403309356797280102nteger )
       => ( ( sgn_sgn_Code_integer(B) != zero_z3403309356797280102nteger )
         => ( sgn_sgn_Code_integer(A) = uminus1351360451143612070nteger(sgn_sgn_Code_integer(B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7185_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) != sgn_sgn_rat(A) )
     => ( ( sgn_sgn_rat(A) != zero_zero_rat )
       => ( ( sgn_sgn_rat(B) != zero_zero_rat )
         => ( sgn_sgn_rat(A) = uminus_uminus_rat(sgn_sgn_rat(B)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_7186_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_7187_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_7188_sgn__minus__1,axiom,
    sgn_sgn_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% sgn_minus_1
tff(fact_7189_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_7190_sgn__minus__1,axiom,
    sgn_sgn_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% sgn_minus_1
tff(fact_7191_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( sgn_sgn_Code_integer(B) = sgn_sgn_Code_integer(A) )
     => ( abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ) ) ).

% same_sgn_abs_add
tff(fact_7192_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% same_sgn_abs_add
tff(fact_7193_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( sgn_sgn_rat(B) = sgn_sgn_rat(A) )
     => ( abs_abs_rat(plus_plus_rat(A,B)) = plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% same_sgn_abs_add
tff(fact_7194_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(A),B)) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% same_sgn_abs_add
tff(fact_7195_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : abs_abs_rat(K) = aa_rat_rat(times_times_rat(K),sgn_sgn_rat(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7196_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_7197_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(times_times_int(K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7198_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K),sgn_sgn_Code_integer(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_7199_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),sgn_sgn_rat(A)) = A ).

% abs_mult_sgn
tff(fact_7200_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_7201_abs__mult__sgn,axiom,
    ! [A: int] : aa_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_7202_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(abs_abs_Code_integer(A)),sgn_sgn_Code_integer(A)) = A ).

% abs_mult_sgn
tff(fact_7203_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_7204_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(A)),abs_abs_rat(A)) = A ).

% sgn_mult_abs
tff(fact_7205_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_7206_sgn__mult__abs,axiom,
    ! [A: int] : aa_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_7207_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(sgn_sgn_Code_integer(A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_7208_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_7209_mult__sgn__abs,axiom,
    ! [Xd: rat] : aa_rat_rat(times_times_rat(sgn_sgn_rat(Xd)),abs_abs_rat(Xd)) = Xd ).

% mult_sgn_abs
tff(fact_7210_mult__sgn__abs,axiom,
    ! [Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xd)),aa_real_real(abs_abs_real,Xd)) = Xd ).

% mult_sgn_abs
tff(fact_7211_mult__sgn__abs,axiom,
    ! [Xd: int] : aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,Xd)),aa_int_int(abs_abs_int,Xd)) = Xd ).

% mult_sgn_abs
tff(fact_7212_mult__sgn__abs,axiom,
    ! [Xd: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(sgn_sgn_Code_integer(Xd)),abs_abs_Code_integer(Xd)) = Xd ).

% mult_sgn_abs
tff(fact_7213_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_7214_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_7215_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( 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_7216_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_7217_sgn__1__pos,axiom,
    ! [A: 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_7218_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_7219_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(sgn_sgn_Code_integer(A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_7220_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_7221_abs__sgn__eq,axiom,
    ! [A: rat] :
      abs_abs_rat(sgn_sgn_rat(A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_7222_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_7223_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_7224_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_7225_sgn__if,axiom,
    ! [Xd: int] :
      aa_int_int(sgn_sgn_int,Xd) = $ite(
        Xd = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xd),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_7226_sgn__if,axiom,
    ! [Xd: real] :
      aa_real_real(sgn_sgn_real,Xd) = $ite(
        Xd = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_7227_sgn__if,axiom,
    ! [Xd: code_integer] :
      sgn_sgn_Code_integer(Xd) = $ite(
        Xd = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xd),one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) ) ).

% sgn_if
tff(fact_7228_sgn__if,axiom,
    ! [Xd: rat] :
      sgn_sgn_rat(Xd) = $ite(
        Xd = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xd),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_if
tff(fact_7229_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_7230_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_7231_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer(A) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_7232_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = 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_7233_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_7234_norm__sgn,axiom,
    ! [Xd: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,Xd)) = $ite(Xd = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7235_norm__sgn,axiom,
    ! [Xd: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,Xd)) = $ite(Xd = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_7236_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_7237_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_7238_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_7239_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xd: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xca: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xd,Xaa),Xba),Xca) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xca,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xd,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xd,Xaa),Xca))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7240_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_7241_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(modulo_modulo_int(K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_7242_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(divide_divide_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),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,M),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_7243_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))))) ) ).

% modulo_int_unfold
tff(fact_7244_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(divide_divide_int(A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A),B)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_7245_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
     => ( ( ? [A4: $o,B2: $o] : A1 = vEBT_Leaf((A4),(B2))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M2)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2) )
                     => ( ( M2 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                         => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_12)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M2)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2) )
                       => ( ( M2 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                           => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_12)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M2)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2) )
                         => ( ( M2 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                             => ( ! [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,numeral_numeral_nat(bit0(one))),M2))
                                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
                               => ( ( ( Mi = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma2)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg2))
                                     => ~ ( ( Mi != Ma2 )
                                         => ! [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,numeral_numeral_nat(bit0(one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I4 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N)) )
                                                   => ( 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),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M2)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2) )
                           => ( ( M2 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                               => ( ! [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,numeral_numeral_nat(bit0(one))),M2))
                                     => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),X_13)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
                                 => ( ( ( Mi = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma2)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg2))
                                       => ~ ( ( Mi != Ma2 )
                                           => ! [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,numeral_numeral_nat(bit0(one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I4 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N)) )
                                                     => ( 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),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_7246_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
    <=> ( ( ? [A6: $o,B6: $o] : A1 = vEBT_Leaf((A6),(B6))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_13) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_13) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma3)),A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ! [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,numeral_numeral_nat(bit0(one))),N6))
               => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),X_13)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I2) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),A22))
            & ( ( Mi2 != Ma3 )
             => ! [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,numeral_numeral_nat(bit0(one))),N6))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N6)) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X5)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma3)),A22,TreeList3),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ! [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,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,N6)))
               => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),X_13)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I2) ) )
            & ( ( Mi2 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),A22))
            & ( ( Mi2 != Ma3 )
             => ! [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,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,N6)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N6)) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X5),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X5),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X5)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_7247_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ).

% and_int_unfold
tff(fact_7248_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),B) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.right_idem
tff(fact_7249_and_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.right_idem
tff(fact_7250_and_Oright__idem,axiom,
    ! [A: code_integer,B: code_integer] : bit_se3949692690581998587nteger(bit_se3949692690581998587nteger(A,B),B) = bit_se3949692690581998587nteger(A,B) ).

% and.right_idem
tff(fact_7251_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(A),B) ).

% and.left_idem
tff(fact_7252_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.left_idem
tff(fact_7253_and_Oleft__idem,axiom,
    ! [A: code_integer,B: code_integer] : bit_se3949692690581998587nteger(A,bit_se3949692690581998587nteger(A,B)) = bit_se3949692690581998587nteger(A,B) ).

% and.left_idem
tff(fact_7254_and_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),A) = A ).

% and.idem
tff(fact_7255_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_7256_and_Oidem,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,A) = A ).

% and.idem
tff(fact_7257_mi__eq__ma__no__ch,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Deg)
     => ( ( Mi3 = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList))
             => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_12) )
          & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_7258_insert__simp__mima,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        | ( Xd = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
       => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_7259_bit_Oconj__zero__right,axiom,
    ! [Xd: int] : aa_int_int(bit_se725231765392027082nd_int(Xd),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_7260_bit_Oconj__zero__right,axiom,
    ! [Xd: code_integer] : bit_se3949692690581998587nteger(Xd,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bit.conj_zero_right
tff(fact_7261_bit_Oconj__zero__left,axiom,
    ! [Xd: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),Xd) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_7262_bit_Oconj__zero__left,axiom,
    ! [Xd: code_integer] : bit_se3949692690581998587nteger(zero_z3403309356797280102nteger,Xd) = zero_z3403309356797280102nteger ).

% bit.conj_zero_left
tff(fact_7263_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_7264_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_7265_zero__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% zero_and_eq
tff(fact_7266_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_7267_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_7268_and__zero__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% and_zero_eq
tff(fact_7269_sgn__le__0__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_7270_zero__le__sgn__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% zero_le_sgn_iff
tff(fact_7271_not__None__eq,axiom,
    ! [Xd: option4927543243414619207at_nat] :
      ( ( Xd != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : Xd = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_7272_not__None__eq,axiom,
    ! [Xd: option_nat] :
      ( ( Xd != none_nat )
    <=> ? [Y5: nat] : Xd = some_nat(Y5) ) ).

% not_None_eq
tff(fact_7273_not__None__eq,axiom,
    ! [Xd: option_num] :
      ( ( Xd != none_num )
    <=> ? [Y5: num] : Xd = some_num(Y5) ) ).

% not_None_eq
tff(fact_7274_not__Some__eq,axiom,
    ! [Xd: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : Xd != some_P7363390416028606310at_nat(Y5)
    <=> ( Xd = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_7275_not__Some__eq,axiom,
    ! [Xd: option_nat] :
      ( ! [Y5: nat] : Xd != some_nat(Y5)
    <=> ( Xd = none_nat ) ) ).

% not_Some_eq
tff(fact_7276_not__Some__eq,axiom,
    ! [Xd: option_num] :
      ( ! [Y5: num] : Xd != some_num(Y5)
    <=> ( Xd = none_num ) ) ).

% not_Some_eq
tff(fact_7277_take__bit__and,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] : bit_se1745604003318907178nteger(Nb,bit_se3949692690581998587nteger(A,B)) = bit_se3949692690581998587nteger(bit_se1745604003318907178nteger(Nb,A),bit_se1745604003318907178nteger(Nb,B)) ).

% take_bit_and
tff(fact_7278_take__bit__and,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_and
tff(fact_7279_take__bit__and,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se725231765392027082nd_int(A),B)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_and
tff(fact_7280_tdeletemimi,axiom,
    ! [Deg: nat,Mi3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Mi3)),Deg,TreeList),Summary)),Xd)),numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))) ) ).

% tdeletemimi
tff(fact_7281_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Mi3)),Deg,TreeList),Summary)),Xd)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_7282_mi__ma__2__deg,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_7283_bit_Oconj__one__right,axiom,
    ! [Xd: int] : aa_int_int(bit_se725231765392027082nd_int(Xd),aa_int_int(uminus_uminus_int,one_one_int)) = Xd ).

% bit.conj_one_right
tff(fact_7284_bit_Oconj__one__right,axiom,
    ! [Xd: code_integer] : bit_se3949692690581998587nteger(Xd,uminus1351360451143612070nteger(one_one_Code_integer)) = Xd ).

% bit.conj_one_right
tff(fact_7285_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_7286_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,uminus1351360451143612070nteger(one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_7287_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_7288_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(uminus1351360451143612070nteger(one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_7289_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ! [X5: num,Y5: num] : V != some_P6201964756284913402um_num(product_Pair_num_num(X5,Y5))
    <=> ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
tff(fact_7290_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X5: int,Y5: int] : V != some_P4184893108420464158nt_int(product_Pair_int_int(X5,Y5))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_7291_not__Some__eq2,axiom,
    ! [V: option4955102875562834416et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc3658429121746597890et_nat] : V != some_P994283844982041979et_nat(produc6017135503889078682et_nat(X5,Y5))
    <=> ( V = none_P7891021755743879167et_nat ) ) ).

% not_Some_eq2
tff(fact_7292_not__Some__eq2,axiom,
    ! [V: option5899113203645146382et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc8799533983703829792et_nat] : V != some_P3006888012174695577et_nat(produc7291855603632349880et_nat(X5,Y5))
    <=> ( V = none_P2758821087663475997et_nat ) ) ).

% not_Some_eq2
tff(fact_7293_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X5: nat,Y5: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y5))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_7294_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_7295_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_7296_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Xd)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))
          | ( Xd = Mi3 )
          | ( Xd = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_7297_less__eq__option__Some__None,axiom,
    ! [Xd: nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,some_nat(Xd)),none_nat) ).

% less_eq_option_Some_None
tff(fact_7298_less__eq__option__Some__None,axiom,
    ! [Xd: num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,some_num(Xd)),none_num) ).

% less_eq_option_Some_None
tff(fact_7299_less__option__Some,axiom,
    ! [Xd: real,Y: real] :
      ( ord_less_option_real(some_real(Xd),some_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% less_option_Some
tff(fact_7300_less__option__Some,axiom,
    ! [Xd: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(Xd),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y) ) ).

% less_option_Some
tff(fact_7301_less__option__Some,axiom,
    ! [Xd: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,some_num(Xd)),some_num(Y))
    <=> aa_num_o(ord_less_num(Xd),Y) ) ).

% less_option_Some
tff(fact_7302_less__option__Some,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,some_nat(Xd)),some_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y) ) ).

% less_option_Some
tff(fact_7303_less__option__Some,axiom,
    ! [Xd: int,Y: int] :
      ( ord_less_option_int(some_int(Xd),some_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Y) ) ).

% less_option_Some
tff(fact_7304_less__option__None__Some__code,axiom,
    ! [Xd: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),some_nat(Xd)) ).

% less_option_None_Some_code
tff(fact_7305_less__option__None__Some__code,axiom,
    ! [Xd: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),some_num(Xd)) ).

% less_option_None_Some_code
tff(fact_7306_member__inv,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Xd)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
        & ( ( Xd = Mi3 )
          | ( Xd = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))) ) ) ) ) ).

% member_inv
tff(fact_7307_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xd: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Xd) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_7308_and__numerals_I8_J,axiom,
    ! [Xd: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xd))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_7309_and__numerals_I8_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_7310_and__numerals_I8_J,axiom,
    ! [Xd: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit1,Xd)),one_one_Code_integer) = one_one_Code_integer ).

% and_numerals(8)
tff(fact_7311_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_7312_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(aa_num_num(bit1,Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_7313_and__numerals_I2_J,axiom,
    ! [Y: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit1,Y))) = one_one_Code_integer ).

% and_numerals(2)
tff(fact_7314_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_num_int(numeral_numeral_int,bit0(Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_7315_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(bit0(Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_7316_and__numerals_I1_J,axiom,
    ! [Y: num] : bit_se3949692690581998587nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(Y))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_7317_and__numerals_I5_J,axiom,
    ! [Xd: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(Xd))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_7318_and__numerals_I5_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(Xd))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_7319_and__numerals_I5_J,axiom,
    ! [Xd: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(bit0(Xd)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_7320_and__numerals_I3_J,axiom,
    ! [Xd: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(Xd))),aa_num_int(numeral_numeral_int,bit0(Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xd)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_7321_and__numerals_I3_J,axiom,
    ! [Xd: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(Xd))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(Xd)),numeral_numeral_nat(Y))) ).

% and_numerals(3)
tff(fact_7322_and__numerals_I3_J,axiom,
    ! [Xd: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(bit0(Xd)),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se3949692690581998587nteger(numera6620942414471956472nteger(Xd),numera6620942414471956472nteger(Y))) ).

% and_numerals(3)
tff(fact_7323_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_7324_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_7325_and__numerals_I4_J,axiom,
    ! [Xd: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(Xd))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xd)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_7326_and__numerals_I4_J,axiom,
    ! [Xd: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(Xd))),numeral_numeral_nat(aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(Xd)),numeral_numeral_nat(Y))) ).

% and_numerals(4)
tff(fact_7327_and__numerals_I4_J,axiom,
    ! [Xd: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(bit0(Xd)),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se3949692690581998587nteger(numera6620942414471956472nteger(Xd),numera6620942414471956472nteger(Y))) ).

% and_numerals(4)
tff(fact_7328_and__numerals_I6_J,axiom,
    ! [Xd: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xd))),aa_num_int(numeral_numeral_int,bit0(Y))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xd)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_7329_and__numerals_I6_J,axiom,
    ! [Xd: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(Xd)),numeral_numeral_nat(Y))) ).

% and_numerals(6)
tff(fact_7330_and__numerals_I6_J,axiom,
    ! [Xd: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit1,Xd)),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se3949692690581998587nteger(numera6620942414471956472nteger(Xd),numera6620942414471956472nteger(Y))) ).

% and_numerals(6)
tff(fact_7331_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,bit0(Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_7332_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,bit0(Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_7333_and__numerals_I7_J,axiom,
    ! [Xd: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xd))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xd)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_7334_and__numerals_I7_J,axiom,
    ! [Xd: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(Xd)),numeral_numeral_nat(Y)))) ).

% and_numerals(7)
tff(fact_7335_and__numerals_I7_J,axiom,
    ! [Xd: num,Y: num] : bit_se3949692690581998587nteger(numera6620942414471956472nteger(aa_num_num(bit1,Xd)),numera6620942414471956472nteger(aa_num_num(bit1,Y))) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),bit_se3949692690581998587nteger(numera6620942414471956472nteger(Xd),numera6620942414471956472nteger(Y)))) ).

% and_numerals(7)
tff(fact_7336_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_7337_of__int__and__eq,axiom,
    ! [K: int,L: int] : ring_18347121197199848620nteger(aa_int_int(bit_se725231765392027082nd_int(K),L)) = bit_se3949692690581998587nteger(ring_18347121197199848620nteger(K),ring_18347121197199848620nteger(L)) ).

% of_int_and_eq
tff(fact_7338_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(B),aa_int_int(bit_se725231765392027082nd_int(A),C)) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.left_commute
tff(fact_7339_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_7340_and_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : bit_se3949692690581998587nteger(B,bit_se3949692690581998587nteger(A,C)) = bit_se3949692690581998587nteger(A,bit_se3949692690581998587nteger(B,C)) ).

% and.left_commute
tff(fact_7341_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(bit_se725231765392027082nd_int(B),A) ).

% and.commute
tff(fact_7342_and_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).

% and.commute
tff(fact_7343_and_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : bit_se3949692690581998587nteger(A,B) = bit_se3949692690581998587nteger(B,A) ).

% and.commute
tff(fact_7344_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B)),C) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B),C)) ).

% and.assoc
tff(fact_7345_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_7346_and_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : bit_se3949692690581998587nteger(bit_se3949692690581998587nteger(A,B),C) = bit_se3949692690581998587nteger(A,bit_se3949692690581998587nteger(B,C)) ).

% and.assoc
tff(fact_7347_prod__decode__aux_Ocases,axiom,
    ! [Xd: product_prod_nat_nat] :
      ~ ! [K2: nat,M2: nat] : Xd != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M2) ).

% prod_decode_aux.cases
tff(fact_7348_of__nat__and__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_and_eq
tff(fact_7349_of__nat__and__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_and_eq
tff(fact_7350_of__nat__and__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb)) = bit_se3949692690581998587nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_and_eq
tff(fact_7351_combine__options__cases,axiom,
    ! [Xd: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xd = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xd),Y) )
       => ( ! [A4: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( Xd = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xd),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7352_combine__options__cases,axiom,
    ! [Xd: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xd = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xd),Y) )
       => ( ! [A4: product_prod_nat_nat,B2: nat] :
              ( ( Xd = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xd),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7353_combine__options__cases,axiom,
    ! [Xd: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xd = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xd),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xd),Y) )
       => ( ! [A4: product_prod_nat_nat,B2: num] :
              ( ( Xd = some_P7363390416028606310at_nat(A4) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xd),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7354_combine__options__cases,axiom,
    ! [Xd: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xd = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xd),Y) )
       => ( ! [A4: nat,B2: product_prod_nat_nat] :
              ( ( Xd = some_nat(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xd),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7355_combine__options__cases,axiom,
    ! [Xd: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xd = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xd),Y) )
       => ( ! [A4: nat,B2: nat] :
              ( ( Xd = some_nat(A4) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xd),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7356_combine__options__cases,axiom,
    ! [Xd: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xd = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xd),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xd),Y) )
       => ( ! [A4: nat,B2: num] :
              ( ( Xd = some_nat(A4) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xd),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7357_combine__options__cases,axiom,
    ! [Xd: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xd = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xd),Y) )
       => ( ! [A4: num,B2: product_prod_nat_nat] :
              ( ( Xd = some_num(A4) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xd),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7358_combine__options__cases,axiom,
    ! [Xd: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xd = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xd),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xd),Y) )
       => ( ! [A4: num,B2: nat] :
              ( ( Xd = some_num(A4) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xd),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7359_combine__options__cases,axiom,
    ! [Xd: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xd = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xd),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xd),Y) )
       => ( ! [A4: num,B2: num] :
              ( ( Xd = some_num(A4) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xd),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xd),Y) ) ) ) ).

% combine_options_cases
tff(fact_7360_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X5: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X5)) ) ) ).

% split_option_all
tff(fact_7361_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X5: nat] : aa_option_nat_o(P,some_nat(X5)) ) ) ).

% split_option_all
tff(fact_7362_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X5: num] : aa_option_num_o(P,some_num(X5)) ) ) ).

% split_option_all
tff(fact_7363_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X5: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X5)) ) ) ).

% split_option_ex
tff(fact_7364_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_13: option_nat] : aa_option_nat_o(P,X_13)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X5: nat] : aa_option_nat_o(P,some_nat(X5)) ) ) ).

% split_option_ex
tff(fact_7365_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X5: num] : aa_option_num_o(P,some_num(X5)) ) ) ).

% split_option_ex
tff(fact_7366_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_7367_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] : Y != some_nat(X22) ) ).

% option.exhaust
tff(fact_7368_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_7369_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_7370_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_7371_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_7372_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_7373_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_7374_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_7375_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xd: vEBT_VEBT] :
      ( ! [A4: $o,B2: $o] : Xd != vEBT_Leaf((A4),(B2))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
       => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_7376_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,Va2),Vb)),Xd)
    <=> ( ( Xd = Mi3 )
        | ( Xd = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_7377_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7378_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( ( A = uminus1351360451143612070nteger(one_one_Code_integer) )
        & ( B = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_7379_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xd),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_7380_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_7381_AND__upper2,axiom,
    ! [Y: int,Xd: 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(Xd),Y)),Y) ) ).

% AND_upper2
tff(fact_7382_AND__upper1,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xd),Y)),Xd) ) ).

% AND_upper1
tff(fact_7383_AND__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(Xd),Y)) ) ).

% AND_lower
tff(fact_7384_real__sgn__eq,axiom,
    ! [Xd: real] : aa_real_real(sgn_sgn_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),aa_real_real(abs_abs_real,Xd)) ).

% real_sgn_eq
tff(fact_7385_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,TreeList),Summary)),Xd) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_7386_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,TreeList),Summary)),Xd) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_7387_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xd),Xd)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary) ).

% vebt_insert.simps(4)
tff(fact_7388_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xd) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_7389_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xd) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_7390_less__option__None__is__Some,axiom,
    ! [Xd: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),Xd)
     => ? [Z2: nat] : Xd = some_nat(Z2) ) ).

% less_option_None_is_Some
tff(fact_7391_less__option__None__is__Some,axiom,
    ! [Xd: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),Xd)
     => ? [Z2: num] : Xd = some_num(Z2) ) ).

% less_option_None_is_Some
tff(fact_7392_less__option__None__Some,axiom,
    ! [Xd: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),some_nat(Xd)) ).

% less_option_None_Some
tff(fact_7393_less__option__None__Some,axiom,
    ! [Xd: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),some_num(Xd)) ).

% less_option_None_Some
tff(fact_7394_height__node,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary))) ) ).

% height_node
tff(fact_7395_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xd: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Xd),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_7396_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_7397_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_7398_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xd: vEBT_VEBT] :
      ( ( Xd != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xd != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xd != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_7399_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xd: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xd) ).

% vebt_member.simps(3)
tff(fact_7400_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),B) ) ) ).

% even_and_iff
tff(fact_7401_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) ) ) ).

% even_and_iff
tff(fact_7402_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),bit_se3949692690581998587nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(bit0(one))),B) ) ) ).

% even_and_iff
tff(fact_7403_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_7404_option_Osize_I4_J,axiom,
    ! [X2: nat] : size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_7405_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_7406_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_7407_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,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,bit0(one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ) ) ).

% even_and_iff_int
tff(fact_7408_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xd: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xd) ).

% vebt_member.simps(4)
tff(fact_7409_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_7410_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_7411_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_7412_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd)),Ve) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_7413_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% one_and_eq
tff(fact_7414_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),numeral_numeral_nat(bit0(one))) ).

% one_and_eq
tff(fact_7415_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% one_and_eq
tff(fact_7416_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,bit0(one))) ).

% and_one_eq
tff(fact_7417_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),numeral_numeral_nat(bit0(one))) ).

% and_one_eq
tff(fact_7418_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% and_one_eq
tff(fact_7419_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xd: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xd )
     => ( ( Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_7420_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_7421_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_7422_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_7423_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_7424_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xd) = numeral_numeral_nat(bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_7425_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% and_int_rec
tff(fact_7426_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xd) = numeral_numeral_nat(bit0(one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_7427_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [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,numeral_numeral_nat(bit0(one))),M))
                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg))
                     => ( ( ( Mi3 != Ma )
                         => ! [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,numeral_numeral_nat(bit0(one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Nb) = I3 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Nb)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Nb) = I3 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_7428_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Nb: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi3: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [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,numeral_numeral_nat(bit0(one))),M))
                   => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),X_13)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
               => ( ( ( Mi3 = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Deg))
                     => ( ( ( Mi3 != Ma )
                         => ! [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,numeral_numeral_nat(bit0(one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Nb) = I3 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),Nb)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Nb) = I3 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_7429_succ__list__to__short,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Xd)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_7430_pred__list__to__short,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi3: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_7431_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,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,numeral_numeral_nat(bit0(one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),numeral_numeral_nat(L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_7432_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Q2)),one_one_int),aa_int_int(minus_minus_int(R2),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_7433_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(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_7434_geqmaxNone,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Xd)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_7435_del__single__cont,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & ( Xd = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ) ) ).

% del_single_cont
tff(fact_7436_power__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xd),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xd)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_7437_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_delete(T,Xd)),Nb) ) ).

% delete_pres_valid
tff(fact_7438_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T,Xd)),Y)
      <=> ( ( Xd != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_7439_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_delete(T,Xd)),Y)
      <=> ( ( Xd != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_7440_pred__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xd),Px) ) ) ).

% pred_corr
tff(fact_7441_succ__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xd),Sx) ) ) ).

% succ_corr
tff(fact_7442_pred__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(T)),Xd),Sx) ) ) ).

% pred_correct
tff(fact_7443_succ__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(T)),Xd),Sx) ) ) ).

% succ_correct
tff(fact_7444_helpyd,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ) ).

% helpyd
tff(fact_7445_helpypredd,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ) ).

% helpypredd
tff(fact_7446_delt__out__of__range,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary) ) ) ) ).

% delt_out_of_range
tff(fact_7447_succ__min,axiom,
    ! [Deg: nat,Xd: nat,Mi3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = some_nat(Mi3) ) ) ) ).

% succ_min
tff(fact_7448_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xd: nat,Mi3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_7449_lesseq__shift,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xd)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_7450_and__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(Xd))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7451_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)),numeral_numeral_nat(bit0(Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_7452_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)),numeral_numeral_nat(aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_7453_and__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7454_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),numeral_numeral_nat(bit0(one))) ).

% Suc_0_and_eq
tff(fact_7455_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),numeral_numeral_nat(bit0(one))) ).

% and_Suc_0_eq
tff(fact_7456_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_7457_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_7458_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_7459_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_7460_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_7461_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_7462_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Va2) = none_nat ).

% vebt_succ.simps(3)
tff(fact_7463_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va2),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_7464_and__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_7465_xor__num_Ocases,axiom,
    ! [Xd: product_prod_num_num] :
      ( ( Xd != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xd != product_Pair_num_num(one,bit0(N))
       => ( ! [N: num] : Xd != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M2: num] : Xd != product_Pair_num_num(bit0(M2),one)
           => ( ! [M2: num,N: num] : Xd != product_Pair_num_num(bit0(M2),bit0(N))
             => ( ! [M2: num,N: num] : Xd != product_Pair_num_num(bit0(M2),aa_num_num(bit1,N))
               => ( ! [M2: num] : Xd != product_Pair_num_num(aa_num_num(bit1,M2),one)
                 => ( ! [M2: num,N: num] : Xd != product_Pair_num_num(aa_num_num(bit1,M2),bit0(N))
                   => ~ ! [M2: num,N: num] : Xd != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_7466_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_7467_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_7468_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B2: $o,X: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),X)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_7469_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_7470_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_7471_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B2: $o] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),zero_zero_nat)
     => ( ! [A4: $o,B2: $o] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: $o,B2: $o,N: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Uu2)
           => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2),X)
             => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),X)
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_7472_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)
     => ( ! [A4: $o,Uw2: $o] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A4: $o,B2: $o,Va: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3),Vb2)
           => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2),Vf2)
             => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Vj2)
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_7473_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B2: $o] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Va3)
         => ( ! [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2),Ve2)
           => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_7474_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B2: $o,X: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),X)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2),X)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2),X)
         => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2),X)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_7475_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B2: $o,X: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),X)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),X)
       => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2),X)
         => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc),X)
           => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_7476_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xd: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xd != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Uz2)
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2),X)
         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc),X)
           => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] : Xd != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_7477_and__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = $ite(
        ( ( M = 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,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))) ) ).

% and_nat_unfold
tff(fact_7478_and__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))) ).

% and_nat_rec
tff(fact_7479_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(T,Xd))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(T),Xd)),numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))) ) ) ).

% minNull_delete_time_bound
tff(fact_7480_mintlistlength,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Nb)
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Ma)
          & ? [M2: nat] :
              ( ( some_nat(M2) = vEBT_vebt_mint(Summary) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_7481_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,Xd: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ).

% vebt_delete.simps(6)
tff(fact_7482_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,Xd: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,TrLst),Smry),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_7483_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1) ) ).

% not_min_Null_member
tff(fact_7484_min__Null__member,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd) ) ).

% min_Null_member
tff(fact_7485_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_7486_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T) ) ).

% minminNull
tff(fact_7487_mint__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Maxi) ) ) ).

% mint_member
tff(fact_7488_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Mini: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xd) ) ) ) ).

% mint_corr_help
tff(fact_7489_mint__sound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xd)
       => ( vEBT_vebt_mint(T) = some_nat(Xd) ) ) ) ).

% mint_sound
tff(fact_7490_mint__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xd) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xd) ) ) ).

% mint_corr
tff(fact_7491_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(T,Xd))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(T),Xd)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_7492_misiz,axiom,
    ! [T: vEBT_VEBT,Nb: nat,M: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ) ) ).

% misiz
tff(fact_7493_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_7494_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_7495_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xd)
     => ( ( Xd != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_7496_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xd)
      <=> (Y) )
     => ( ( ( Xd = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xd = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xd = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_7497_insertsimp,axiom,
    ! [T: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),L)),numeral_numeral_nat(aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_7498_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((A),(B)) ).

% vebt_delete.simps(3)
tff(fact_7499_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),zero_zero_nat) = vEBT_Leaf($false,(B)) ).

% vebt_delete.simps(1)
tff(fact_7500_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ).

% vebt_delete.simps(4)
tff(fact_7501_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ).

% vebt_delete.simps(2)
tff(fact_7502_nested__mint,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Nb: nat,Va2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Mi3)
         => ( ( Ma != Mi3 )
           => 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(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),numeral_numeral_nat(bit0(one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),numeral_numeral_nat(bit0(one)))))),size_s6755466524823107622T_VEBT(TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_7503_vebt__mint_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xd) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( Y != $ite(
                  (A4),
                  some_nat(zero_zero_nat),
                  $ite((B2),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_7504_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_7505_greater__shift,axiom,
    ! [Y: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xd)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xd)),some_nat(Y)) ) ).

% greater_shift
tff(fact_7506_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_7507_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_7508_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_7509_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_7510_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_7511_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_7512_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_7513_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_7514_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_7515_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_7516_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_7517_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_7518_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_7519_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_7520_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_7521_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xd: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xd,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V3: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V3)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A4) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B2) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xd,A4),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_7522_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xd: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xd,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V3: num] : Xaa = some_num(V3)
           => ( ( Xba = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xaa = some_num(A4) )
               => ! [B2: num] :
                    ( ( Xba = some_num(B2) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xd,A4),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_7523_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xd: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xd),Xaa),Xba) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V3: nat] : Xaa = some_nat(V3)
           => ( ( Xba = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xaa = some_nat(A4) )
               => ! [B2: nat] :
                    ( ( Xba = some_nat(B2) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xd,A4),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_7524_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xd: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : Xd != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V3: product_prod_nat_nat] : Xd != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V3),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2028404973800837934_nat_o,X: product_prod_nat_nat,Y2: product_prod_nat_nat] : Xd != produc3653971535690468304at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X),some_P7363390416028606310at_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_7525_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xd: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : Xd != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V3: nat] : Xd != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V3),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_o,X: nat,Y2: nat] : Xd != produc7967433646104525454on_nat(F3,produc5098337634421038937on_nat(some_nat(X),some_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_7526_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xd: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : Xd != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V3: num] : Xd != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(some_num(V3),none_num))
       => ~ ! [F3: fun_num_fun_num_o,X: num,Y2: num] : Xd != produc1140783415258738358on_num(F3,produc8585076106096196333on_num(some_num(X),some_num(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_7527_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xd: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : Xd != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V3: product_prod_nat_nat] : Xd != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V3),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B2: product_prod_nat_nat] : Xd != produc7814092408141220007at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_7528_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xd: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : Xd != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V3: nat] : Xd != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V3),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_nat,A4: nat,B2: nat] : Xd != produc6699827698519632094on_nat(F3,produc5098337634421038937on_nat(some_nat(A4),some_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_7529_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xd: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : Xd != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V3: num] : Xd != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V3),none_num))
       => ~ ! [F3: fun_num_fun_num_num,A4: num,B2: num] : Xd != produc6360833703471310868on_num(F3,produc8585076106096196333on_num(some_num(A4),some_num(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_7530_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_7531_summaxma,axiom,
    ! [Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary)),Deg)
     => ( ( Mi3 != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) ) ) ) ).

% summaxma
tff(fact_7532_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_7533_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_7534_mul__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xd),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xd)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_7535_maxbmo,axiom,
    ! [T: vEBT_VEBT,Xd: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(Xd) )
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xd) ) ).

% maxbmo
tff(fact_7536_add__shift,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xd)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_7537_maxt__member,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Maxi) ) ) ).

% maxt_member
tff(fact_7538_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xd: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_7539_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xd: vEBT_VEBTi,Y: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd),I,Y) = list_u6098035379799741383_VEBTi(Xs,I,Y) ).

% list_update_overwrite
tff(fact_7540_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Maxi: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xd)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_7541_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xd)
       => ( vEBT_vebt_maxt(T) = some_nat(Xd) ) ) ) ).

% maxt_sound
tff(fact_7542_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xd) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xd) ) ) ).

% maxt_corr
tff(fact_7543_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xd: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_7544_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xd: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_7545_length__list__update,axiom,
    ! [Xs: list_real,I: nat,Xd: real] : size_size_list_real(list_update_real(Xs,I,Xd)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_7546_length__list__update,axiom,
    ! [Xs: list_o,I: nat,Xd: $o] : size_size_list_o(list_update_o(Xs,I,(Xd))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_7547_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xd: nat] : size_size_list_nat(list_update_nat(Xs,I,Xd)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_7548_length__list__update,axiom,
    ! [Xs: list_int,I: nat,Xd: int] : size_size_list_int(list_update_int(Xs,I,Xd)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_7549_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_7550_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_7551_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_7552_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_7553_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Xd: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xd)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_7554_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,Xd: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xd)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_7555_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_7556_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi] :
      ( ( I != J )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd),J) = nth_VEBT_VEBTi(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_7557_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_7558_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I)
     => ( list_u6098035379799741383_VEBTi(Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_7559_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),I)
     => ( list_update_real(Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_7560_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xd: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xd)) = Xs ) ) ).

% list_update_beyond
tff(fact_7561_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_7562_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
     => ( list_update_int(Xs,I,Xd) = Xs ) ) ).

% list_update_beyond
tff(fact_7563_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,Xd: vEBT_VEBT] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J,Xd),I) = nth_VEBT_VEBT(L,I) ) ) ).

% nth_update_invalid
tff(fact_7564_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,Xd: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,Xd),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_7565_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,Xd: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_nat_real(nth_real(list_update_real(L,J,Xd)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_7566_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,Xd: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_nat_o(nth_o(list_update_o(L,J,(Xd))),I)
      <=> aa_nat_o(nth_o(L),I) ) ) ).

% nth_update_invalid
tff(fact_7567_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,Xd: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,Xd)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_7568_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,Xd: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_nat_int(nth_int(list_update_int(L,J,Xd)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_7569_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_7570_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_7571_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xd)),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_7572_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xd: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xd))),I)
      <=> (Xd) ) ) ).

% nth_list_update_eq
tff(fact_7573_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xd)),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_7574_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xd)),I) = Xd ) ) ).

% nth_list_update_eq
tff(fact_7575_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_7576_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J)),J,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_7577_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_7578_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_7579_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_7580_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_7581_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                  $ite(Xd = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_7582_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_7583_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I5 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd),I5,X6) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I5,X6),I,Xd) ) ) ).

% list_update_swap
tff(fact_7584_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi,X6: vEBT_VEBTi] :
      ( ( I != I5 )
     => ( list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd),I5,X6) = list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I5,X6),I,Xd) ) ) ).

% list_update_swap
tff(fact_7585_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,Xd: complex,I: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A2)
     => ( aa_set_complex_o(member_complex(Xd),A2)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xd))),A2) ) ) ).

% set_update_subsetI
tff(fact_7586_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xd: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A2)
     => ( aa_set_real_o(member_real(Xd),A2)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xd))),A2) ) ) ).

% set_update_subsetI
tff(fact_7587_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xd: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A2)
     => ( aa_set_int_o(member_int(Xd),A2)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xd))),A2) ) ) ).

% set_update_subsetI
tff(fact_7588_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xd: vEBT_VEBT,I: nat] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xd))),A2) ) ) ).

% set_update_subsetI
tff(fact_7589_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,Xd: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A2)
     => ( member_VEBT_VEBTi(Xd,A2)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xd)),A2) ) ) ).

% set_update_subsetI
tff(fact_7590_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xd: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(Xd),A2)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xd))),A2) ) ) ).

% set_update_subsetI
tff(fact_7591_in__set__upd__eq,axiom,
    ! [I: nat,L: list_complex,Xd: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
     => ( aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( aa_set_complex_o(member_complex(Xd),set_complex2(L))
            & ! [Y5: complex] : aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7592_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xd: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(L))
            & ! [Y5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7593_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xd: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(L))
            & ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7594_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,Xd: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xd),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( aa_set_real_o(member_real(Xd),set_real2(L))
            & ! [Y5: real] : aa_set_real_o(member_real(Xd),set_real2(list_update_real(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7595_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,Xd: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xd),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xd)
          <=> (Y) )
          | ( member_o((Xd),set_o2(L))
            & ! [Y5: $o] : member_o((Xd),set_o2(list_update_o(L,I,(Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7596_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( aa_set_nat_o(member_nat(Xd),set_nat2(L))
            & ! [Y5: nat] : aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7597_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,Xd: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xd),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xd = Y )
          | ( aa_set_int_o(member_int(Xd),set_int2(L))
            & ! [Y5: int] : aa_set_int_o(member_int(Xd),set_int2(list_update_int(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_7598_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xd: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7599_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7600_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7601_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xd),set_real2(list_update_real(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7602_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xd: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o((Xd),set_o2(list_update_o(Xs,Nb,(Xd)))) ) ).

% set_update_memI
tff(fact_7603_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7604_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xd),set_int2(list_update_int(Xs,Nb,Xd))) ) ).

% set_update_memI
tff(fact_7605_in__set__upd__cases,axiom,
    ! [Xd: complex,L: list_complex,I: nat,Y: complex] :
      ( aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
         => ( Xd != Y ) )
       => aa_set_complex_o(member_complex(Xd),set_complex2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7606_in__set__upd__cases,axiom,
    ! [Xd: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
         => ( Xd != Y ) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7607_in__set__upd__cases,axiom,
    ! [Xd: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
         => ( Xd != Y ) )
       => member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7608_in__set__upd__cases,axiom,
    ! [Xd: real,L: list_real,I: nat,Y: real] :
      ( aa_set_real_o(member_real(Xd),set_real2(list_update_real(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( Xd != Y ) )
       => aa_set_real_o(member_real(Xd),set_real2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7609_in__set__upd__cases,axiom,
    ! [Xd: $o,L: list_o,I: nat,Y: $o] :
      ( member_o((Xd),set_o2(list_update_o(L,I,(Y))))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( (Xd)
          <=> ~ (Y) ) )
       => member_o((Xd),set_o2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7610_in__set__upd__cases,axiom,
    ! [Xd: nat,L: list_nat,I: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
         => ( Xd != Y ) )
       => aa_set_nat_o(member_nat(Xd),set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7611_in__set__upd__cases,axiom,
    ! [Xd: int,L: list_int,I: nat,Y: int] :
      ( aa_set_int_o(member_int(Xd),set_int2(list_update_int(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( Xd != Y ) )
       => aa_set_int_o(member_int(Xd),set_int2(L)) ) ) ).

% in_set_upd_cases
tff(fact_7612_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_complex,Xd: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
     => ( aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: complex] : aa_set_complex_o(member_complex(Xd),set_complex2(list_update_complex(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7613_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xd: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7614_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xd: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7615_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,Xd: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xd),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: real] : aa_set_real_o(member_real(Xd),set_real2(list_update_real(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7616_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,Xd: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xd),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xd)
          <=> (Y) )
          | ! [Y5: $o] : member_o((Xd),set_o2(list_update_o(L,I,(Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7617_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: nat] : aa_set_nat_o(member_nat(Xd),set_nat2(list_update_nat(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7618_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,Xd: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xd),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xd = Y )
          | ! [Y5: int] : aa_set_int_o(member_int(Xd),set_int2(list_update_int(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_7619_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xd) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_7620_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,Xd) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_7621_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( ( list_update_real(Xs,I,Xd) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_7622_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xd: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xd)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xd) ) ) ) ).

% list_update_same_conv
tff(fact_7623_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xd) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_7624_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xd) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xd ) ) ) ).

% list_update_same_conv
tff(fact_7625_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xd: vEBT_VEBT,J: nat] :
      nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xd),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)) ),
        Xd,
        nth_VEBT_VEBT(L,J) ) ).

% nth_list_update'
tff(fact_7626_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xd: vEBT_VEBTi,J: nat] :
      nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xd),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)) ),
        Xd,
        nth_VEBT_VEBTi(L,J) ) ).

% nth_list_update'
tff(fact_7627_nth__list__update_H,axiom,
    ! [L: list_real,I: nat,Xd: real,J: nat] :
      aa_nat_real(nth_real(list_update_real(L,I,Xd)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L)) ),
        Xd,
        aa_nat_real(nth_real(L),J) ) ).

% nth_list_update'
tff(fact_7628_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,Xd: $o,J: nat] :
      ( aa_nat_o(nth_o(list_update_o(L,I,(Xd))),J)
    <=> $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L)) ),
          (Xd),
          aa_nat_o(nth_o(L),J) ) ) ).

% nth_list_update'
tff(fact_7629_nth__list__update_H,axiom,
    ! [L: list_nat,I: nat,Xd: nat,J: nat] :
      aa_nat_nat(nth_nat(list_update_nat(L,I,Xd)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L)) ),
        Xd,
        aa_nat_nat(nth_nat(L),J) ) ).

% nth_list_update'
tff(fact_7630_nth__list__update_H,axiom,
    ! [L: list_int,I: nat,Xd: int,J: nat] :
      aa_nat_int(nth_int(list_update_int(L,I,Xd)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L)) ),
        Xd,
        aa_nat_int(nth_int(L),J) ) ).

% nth_list_update'
tff(fact_7631_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xd: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xd),J) = $ite(I = J,Xd,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_7632_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xd: vEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xd),J) = $ite(I = J,Xd,nth_VEBT_VEBTi(Xs,J)) ) ) ).

% nth_list_update
tff(fact_7633_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,Xd: real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xd)),J) = $ite(I = J,Xd,aa_nat_real(nth_real(Xs),J)) ) ) ).

% nth_list_update
tff(fact_7634_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xd: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xd))),J)
      <=> $ite(I = J,(Xd),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_7635_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xd: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xd)),J) = $ite(I = J,Xd,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_7636_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xd: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xd)),J) = $ite(I = J,Xd,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_7637_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_7638_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_7639_vebt__maxt_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xd) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( Y != $ite(
                  (B2),
                  some_nat(one_one_nat),
                  $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_7640_insert__simp__norm,axiom,
    ! [Xd: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( Xd != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_7641_insert__simp__excp,axiom,
    ! [Mi3: nat,Deg: nat,TreeList: list_VEBT_VEBT,Xd: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( Xd != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xd),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi3),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_7642_less__shift,axiom,
    ! [Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xd)),some_nat(Y)) ) ).

% less_shift
tff(fact_7643_max__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)) ).

% max_Suc_Suc
tff(fact_7644_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_7645_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_7646_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_7647_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_7648_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_7649_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_7650_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_or_iff
tff(fact_7651_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_int_int(ord_max_int(aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_7652_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        | (Q) )) = ord_max_Code_integer(zero_n356916108424825756nteger((P)),zero_n356916108424825756nteger((Q))) ).

% of_bool_or_iff
tff(fact_7653_max__0__1_I3_J,axiom,
    ! [Xd: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(Xd)) = numera1916890842035813515d_enat(Xd) ).

% max_0_1(3)
tff(fact_7654_max__0__1_I3_J,axiom,
    ! [Xd: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(Xd)) = numeral_numeral_real(Xd) ).

% max_0_1(3)
tff(fact_7655_max__0__1_I3_J,axiom,
    ! [Xd: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(Xd)) = numeral_numeral_rat(Xd) ).

% max_0_1(3)
tff(fact_7656_max__0__1_I3_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),numeral_numeral_nat(Xd)) = numeral_numeral_nat(Xd) ).

% max_0_1(3)
tff(fact_7657_max__0__1_I3_J,axiom,
    ! [Xd: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,Xd)) = aa_num_int(numeral_numeral_int,Xd) ).

% max_0_1(3)
tff(fact_7658_max__0__1_I4_J,axiom,
    ! [Xd: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xd),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xd) ).

% max_0_1(4)
tff(fact_7659_max__0__1_I4_J,axiom,
    ! [Xd: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xd)),zero_zero_real) = numeral_numeral_real(Xd) ).

% max_0_1(4)
tff(fact_7660_max__0__1_I4_J,axiom,
    ! [Xd: num] : ord_max_rat(numeral_numeral_rat(Xd),zero_zero_rat) = numeral_numeral_rat(Xd) ).

% max_0_1(4)
tff(fact_7661_max__0__1_I4_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(Xd)),zero_zero_nat) = numeral_numeral_nat(Xd) ).

% max_0_1(4)
tff(fact_7662_max__0__1_I4_J,axiom,
    ! [Xd: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xd)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xd) ).

% max_0_1(4)
tff(fact_7663_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_7664_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)),numeral_numeral_real(V),numeral_numeral_real(U)) ).

% max_number_of(1)
tff(fact_7665_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)),numeral_numeral_rat(V),numeral_numeral_rat(U)) ).

% max_number_of(1)
tff(fact_7666_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(U)),numeral_numeral_nat(V)),numeral_numeral_nat(V),numeral_numeral_nat(U)) ).

% max_number_of(1)
tff(fact_7667_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_7668_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_7669_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_7670_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_7671_max__0__1_I2_J,axiom,
    ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_7672_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_7673_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_7674_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_7675_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_7676_max__0__1_I1_J,axiom,
    ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_7677_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_7678_max__0__1_I5_J,axiom,
    ! [Xd: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xd)) = numera1916890842035813515d_enat(Xd) ).

% max_0_1(5)
tff(fact_7679_max__0__1_I5_J,axiom,
    ! [Xd: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(Xd)) = numeral_numeral_real(Xd) ).

% max_0_1(5)
tff(fact_7680_max__0__1_I5_J,axiom,
    ! [Xd: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(Xd)) = numeral_numeral_rat(Xd) ).

% max_0_1(5)
tff(fact_7681_max__0__1_I5_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),numeral_numeral_nat(Xd)) = numeral_numeral_nat(Xd) ).

% max_0_1(5)
tff(fact_7682_max__0__1_I5_J,axiom,
    ! [Xd: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,Xd)) = aa_num_int(numeral_numeral_int,Xd) ).

% max_0_1(5)
tff(fact_7683_max__0__1_I6_J,axiom,
    ! [Xd: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xd),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xd) ).

% max_0_1(6)
tff(fact_7684_max__0__1_I6_J,axiom,
    ! [Xd: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xd)),one_one_real) = numeral_numeral_real(Xd) ).

% max_0_1(6)
tff(fact_7685_max__0__1_I6_J,axiom,
    ! [Xd: num] : ord_max_rat(numeral_numeral_rat(Xd),one_one_rat) = numeral_numeral_rat(Xd) ).

% max_0_1(6)
tff(fact_7686_max__0__1_I6_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(Xd)),one_one_nat) = numeral_numeral_nat(Xd) ).

% max_0_1(6)
tff(fact_7687_max__0__1_I6_J,axiom,
    ! [Xd: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xd)),one_one_int) = aa_num_int(numeral_numeral_int,Xd) ).

% max_0_1(6)
tff(fact_7688_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(4)
tff(fact_7689_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),uminus1351360451143612070nteger(numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_7690_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(4)
tff(fact_7691_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_7692_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)),numeral_numeral_real(V),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(3)
tff(fact_7693_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),uminus1351360451143612070nteger(numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_7694_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)),numeral_numeral_rat(V),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(3)
tff(fact_7695_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_7696_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),numeral_numeral_real(U)) ).

% max_number_of(2)
tff(fact_7697_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_7698_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),numeral_numeral_rat(U)) ).

% max_number_of(2)
tff(fact_7699_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_7700_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_7701_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_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,pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_7702_of__nat__max,axiom,
    ! [Xd: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)) = ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(Xd),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_7703_of__nat__max,axiom,
    ! [Xd: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xd)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_7704_of__nat__max,axiom,
    ! [Xd: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,Xd)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_7705_of__nat__max,axiom,
    ! [Xd: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xd)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_7706_max__diff__distrib__left,axiom,
    ! [Xd: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xd),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_7707_max__diff__distrib__left,axiom,
    ! [Xd: int,Y: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(ord_max_int(Xd),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(minus_minus_int(Xd),Z)),aa_int_int(minus_minus_int(Y),Z)) ).

% max_diff_distrib_left
tff(fact_7708_max__add__distrib__left,axiom,
    ! [Xd: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xd),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_7709_max__add__distrib__left,axiom,
    ! [Xd: rat,Y: rat,Z: rat] : plus_plus_rat(ord_max_rat(Xd,Y),Z) = ord_max_rat(plus_plus_rat(Xd,Z),plus_plus_rat(Y,Z)) ).

% max_add_distrib_left
tff(fact_7710_max__add__distrib__left,axiom,
    ! [Xd: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_7711_max__add__distrib__left,axiom,
    ! [Xd: int,Y: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(ord_max_int(Xd),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xd),Z)),aa_int_int(plus_plus_int(Y),Z)) ).

% max_add_distrib_left
tff(fact_7712_max__add__distrib__right,axiom,
    ! [Xd: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Z)) ).

% max_add_distrib_right
tff(fact_7713_max__add__distrib__right,axiom,
    ! [Xd: rat,Y: rat,Z: rat] : plus_plus_rat(Xd,ord_max_rat(Y,Z)) = ord_max_rat(plus_plus_rat(Xd,Y),plus_plus_rat(Xd,Z)) ).

% max_add_distrib_right
tff(fact_7714_max__add__distrib__right,axiom,
    ! [Xd: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xd),Z)) ).

% max_add_distrib_right
tff(fact_7715_max__add__distrib__right,axiom,
    ! [Xd: int,Y: int,Z: int] : aa_int_int(plus_plus_int(Xd),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xd),Y)),aa_int_int(plus_plus_int(Xd),Z)) ).

% max_add_distrib_right
tff(fact_7716_nat__mult__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),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,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_7717_nat__mult__max__left,axiom,
    ! [M: 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,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_7718_nat__add__max__left,axiom,
    ! [M: 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,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_7719_nat__add__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),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,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_7720_nat__minus__add__max,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),M) ).

% nat_minus_add_max
tff(fact_7721_max__less__iff__conj,axiom,
    ! [Xd: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(Xd,Y)),Z)
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xd),Z)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7722_max__less__iff__conj,axiom,
    ! [Xd: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xd),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7723_max__less__iff__conj,axiom,
    ! [Xd: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(Xd,Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7724_max__less__iff__conj,axiom,
    ! [Xd: num,Y: num,Z: num] :
      ( aa_num_o(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,Xd),Y)),Z)
    <=> ( aa_num_o(ord_less_num(Xd),Z)
        & aa_num_o(ord_less_num(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7725_max__less__iff__conj,axiom,
    ! [Xd: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7726_max__less__iff__conj,axiom,
    ! [Xd: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xd),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_7727_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
     => ( ord_ma741700101516333627d_enat(A,B) = B ) ) ).

% max.absorb4
tff(fact_7728_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_7729_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb4
tff(fact_7730_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(ord_less_num(A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_7731_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_7732_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_7733_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
     => ( ord_ma741700101516333627d_enat(A,B) = A ) ) ).

% max.absorb3
tff(fact_7734_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_7735_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb3
tff(fact_7736_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_7737_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_7738_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_7739_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Xd)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_7740_succ__less__length__list,axiom,
    ! [Deg: nat,Mi3: nat,Xd: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Xd)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_7741_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T)) ).

% set_vebt'_def
tff(fact_7742_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(Q2,zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_7743_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_7744_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_af(A),B))) ).

% finite_interval_int4
tff(fact_7745_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ag(Nb),C))) ) ).

% finite_nth_roots
tff(fact_7746_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ah(A),B))) ).

% finite_interval_int3
tff(fact_7747_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ai(A),B))) ).

% finite_interval_int2
tff(fact_7748_del__x__not__mi,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                   => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                    $ite(
                                      Xd = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi3,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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                  $ite(Xd = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_7749_del__x__not__mi__new__node__nil,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                    $ite(
                                      Xd = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Mi3,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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_7750_del__x__not__mia,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
               => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                    $ite(
                                      Xd = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi3,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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),
                                  $ite(Xd = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg,newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_7751_del__x__mia,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),
                $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,Deg),numeral_numeral_nat(bit0(one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(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,Deg),numeral_numeral_nat(bit0(one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg,newlist),Summary) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_7752_del__x__mi__lets__in__minNull,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(Sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_7753_del__x__mi__lets__in,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xd = Mi3 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg,Newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_7754_del__x__mi,axiom,
    ! [Xd: nat,Mi3: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( Xd = Mi3 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg,newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_7755_del__in__range,axiom,
    ! [Mi3: nat,Xd: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Xd)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma) )
     => ( ( Mi3 != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                xn: nat,
                xn:= 
                  $ite(Xd = Mi3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xd),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xd = Mi3,xn,Mi3),
                  $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,Deg),numeral_numeral_nat(bit0(one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(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,Deg),numeral_numeral_nat(bit0(one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,h),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                      $ite(
                                        xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg,newlist),Summary) ) ) ),
                      aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_7756_pred__lesseq__max,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,Mi3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd),some_nat(Mi3),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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_7757_pred__less__length__list,axiom,
    ! [Deg: nat,Xd: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi3: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xd),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Deg,TreeList),Summary),Xd) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd),some_nat(Mi3),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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_7758_max__def__raw,axiom,
    ! [X4: extended_enat,Xa: extended_enat] :
      ord_ma741700101516333627d_enat(X4,Xa) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7759_max__def__raw,axiom,
    ! [X4: set_nat,Xa: set_nat] :
      ord_max_set_nat(X4,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7760_max__def__raw,axiom,
    ! [X4: rat,Xa: rat] :
      ord_max_rat(X4,Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7761_max__def__raw,axiom,
    ! [X4: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = $ite(aa_num_o(ord_less_eq_num(X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7762_max__def__raw,axiom,
    ! [X4: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7763_max__def__raw,axiom,
    ! [X4: int,Xa: int] :
      aa_int_int(ord_max_int(X4),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_7764_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(bit0(Nb)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m),m) ) ).

% numeral_code(2)
tff(fact_7765_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(bit0(Nb)) = $let(
        m: real,
        m:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_7766_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(bit0(Nb)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Nb),
        plus_plus_rat(m,m) ) ).

% numeral_code(2)
tff(fact_7767_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_nat(bit0(Nb)) = $let(
        m: nat,
        m:= numeral_numeral_nat(Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_7768_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,bit0(Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(m),m) ) ).

% numeral_code(2)
tff(fact_7769_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_7770_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aTP_Lamm_aj(A))),collect_complex(aTP_Lamm_aj(B)))
    <=> ( aa_complex_o(dvd_dvd_complex(A),B)
        & ~ aa_complex_o(dvd_dvd_complex(B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7771_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7772_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_al,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_al,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_7773_lambda__one,axiom,
    aTP_Lamm_am = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_7774_lambda__one,axiom,
    aTP_Lamm_an = times_times_assn(one_one_assn) ).

% lambda_one
tff(fact_7775_lambda__one,axiom,
    aTP_Lamm_ao = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_7776_lambda__one,axiom,
    aTP_Lamm_ap = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_7777_lambda__one,axiom,
    aTP_Lamm_aq = times_times_int(one_one_int) ).

% lambda_one
tff(fact_7778_lambda__one,axiom,
    aTP_Lamm_ar = times_3573771949741848930nteger(one_one_Code_integer) ).

% lambda_one
tff(fact_7779_lambda__one,axiom,
    aTP_Lamm_as = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_7780_lambda__zero,axiom,
    aTP_Lamm_at = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_7781_lambda__zero,axiom,
    aTP_Lamm_au = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_7782_lambda__zero,axiom,
    aTP_Lamm_av = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_7783_lambda__zero,axiom,
    aTP_Lamm_aw = times_times_int(zero_zero_int) ).

% lambda_zero
tff(fact_7784_lambda__zero,axiom,
    aTP_Lamm_ax = times_3573771949741848930nteger(zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_7785_lambda__zero,axiom,
    aTP_Lamm_ay = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_7786_mult__commute__abs,axiom,
    ! [C: assn] : aTP_Lamm_az(C) = times_times_assn(C) ).

% mult_commute_abs
tff(fact_7787_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_ba,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_7788_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bb,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_7789_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_bc(C) = times_times_int(C) ).

% mult_commute_abs
tff(fact_7790_mult__commute__abs,axiom,
    ! [C: code_integer] : aTP_Lamm_bd(C) = times_3573771949741848930nteger(C) ).

% mult_commute_abs
tff(fact_7791_mult__commute__abs,axiom,
    ! [C: complex] : aTP_Lamm_be(C) = times_times_complex(C) ).

% mult_commute_abs
tff(fact_7792_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aTP_Lamm_aj(A))),collect_complex(aTP_Lamm_aj(B)))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% subset_divisors_dvd
tff(fact_7793_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_al,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_al,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_7794_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_7795_less__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
    <=> ord_less_fun_nat_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B3)) ) ).

% less_set_def
tff(fact_7796_less__set__def,axiom,
    ! [A2: set_real,B3: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),B3)
    <=> ord_less_fun_real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B3)) ) ).

% less_set_def
tff(fact_7797_less__set__def,axiom,
    ! [A2: set_int,B3: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),B3)
    <=> ord_less_fun_int_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B3)) ) ).

% less_set_def
tff(fact_7798_less__set__def,axiom,
    ! [A2: set_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B3)
    <=> ord_le3534172602983796987VEBT_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B3)) ) ).

% less_set_def
tff(fact_7799_less__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),B3)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B3)) ) ).

% less_set_def
tff(fact_7800_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_al,I))) ) ).

% finite_divisors_int
tff(fact_7801_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bf(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_7802_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_7803_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_7804_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m),m)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_7805_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit1,Nb)) = $let(
        m: real,
        m:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_7806_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(aa_num_num(bit1,Nb)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Nb),
        plus_plus_rat(plus_plus_rat(m,m),one_one_rat) ) ).

% numeral_code(3)
tff(fact_7807_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_nat(aa_num_num(bit1,Nb)) = $let(
        m: nat,
        m:= 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,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_7808_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_7809_power__numeral__even,axiom,
    ! [Z: assn,W: num] :
      aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(bit0(W))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
        aa_assn_assn(times_times_assn(w),w) ) ).

% power_numeral_even
tff(fact_7810_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),numeral_numeral_nat(bit0(W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),numeral_numeral_nat(W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_7811_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(bit0(W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_7812_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),numeral_numeral_nat(bit0(W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),numeral_numeral_nat(W)),
        aa_int_int(times_times_int(w),w) ) ).

% power_numeral_even
tff(fact_7813_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(bit0(W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
        aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(w),w) ) ).

% power_numeral_even
tff(fact_7814_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),numeral_numeral_nat(bit0(W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),numeral_numeral_nat(W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_7815_power__numeral__odd,axiom,
    ! [Z: assn,W: num] :
      aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),numeral_numeral_nat(W)),
        aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7816_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),numeral_numeral_nat(W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7817_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),numeral_numeral_nat(W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7818_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),numeral_numeral_nat(W)),
        aa_int_int(times_times_int(aa_int_int(times_times_int(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7819_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),numeral_numeral_nat(W)),
        aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7820_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),numeral_numeral_nat(aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),numeral_numeral_nat(W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_7821_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(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_7822_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,M))) ) ).

% finite_divisors_nat
tff(fact_7823_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(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_7824_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(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_7825_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(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_7826_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(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_7827_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_bg,Nb))) ) ).

% finite_roots_unity
tff(fact_7828_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bh(Nb))) ) ).

% finite_roots_unity
tff(fact_7829_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bi(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7830_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bj(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7831_finite__lists__length__eq,axiom,
    ! [A2: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_bk(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7832_finite__lists__length__eq,axiom,
    ! [A2: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_bl(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7833_finite__lists__length__eq,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bm(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7834_finite__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bn(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7835_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bo(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7836_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(minus_minus_int(Z),Z4),
          $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_7837_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bp(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7838_finite__lists__length__le,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bq(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7839_finite__lists__length__le,axiom,
    ! [A2: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_br(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7840_finite__lists__length__le,axiom,
    ! [A2: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_bs(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7841_finite__lists__length__le,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bt(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7842_finite__lists__length__le,axiom,
    ! [A2: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bu(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7843_finite__lists__length__le,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bv(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_7844_set__decode__def,axiom,
    ! [Xd: nat] : aa_nat_set_nat(nat_set_decode,Xd) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bw,Xd)) ).

% set_decode_def
tff(fact_7845_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,set_fo1949268297981939178at_rat(aTP_Lamm_bx(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_7846_pochhammer__code,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,set_fo1084959871951514735nteger(aTP_Lamm_by(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_7847_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_bz(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_7848_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_ca(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_7849_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_cb(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_7850_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_cc(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_7851_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,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
          aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
          aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(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_7852_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_7853_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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))),numeral_numeral_nat(bit0(one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_7854_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList),S)),Xd)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_7855_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xd) = Y )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y != numeral_numeral_nat(aa_num_num(bit1,one)) ) )
       => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != numeral_numeral_nat(aa_num_num(bit1,one)) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_7856_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xd) = Y )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y != numeral_numeral_nat(bit0(bit0(one))) ) )
       => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != numeral_numeral_nat(bit0(bit0(one))) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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))),numeral_numeral_nat(bit0(one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_7857_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd)),Xd)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_7858_vebt__buildup_Oelims,axiom,
    ! [Xd: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xd) = Y )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
                        aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
                        aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_7859_vebt__member_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd)
    <=> $ite(
          Xd = Mi3,
          $true,
          $ite(
            Xd = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_7860_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi3: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,V),TreeList),Vc2)),Xd)
    <=> ( ( Xd = Mi3 )
        | ( Xd = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_7861_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2)
               => $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)),numeral_numeral_nat(bit0(one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_7862_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2)
             => ~ $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)),numeral_numeral_nat(bit0(one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_7863_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2)
               => ( (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)),numeral_numeral_nat(bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_7864_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xd: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),ord_ma741700101516333627d_enat(Xd,Y))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Xd)
        | aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7865_less__max__iff__disj,axiom,
    ! [Z: real,Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xd),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xd)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7866_less__max__iff__disj,axiom,
    ! [Z: rat,Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),ord_max_rat(Xd,Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xd)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7867_less__max__iff__disj,axiom,
    ! [Z: num,Xd: num,Y: num] :
      ( aa_num_o(ord_less_num(Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xd),Y))
    <=> ( aa_num_o(ord_less_num(Z),Xd)
        | aa_num_o(ord_less_num(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7868_less__max__iff__disj,axiom,
    ! [Z: nat,Xd: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xd),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xd)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7869_less__max__iff__disj,axiom,
    ! [Z: int,Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xd),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xd)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_7870_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(B,C)),A)
     => ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
         => ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A) ) ) ).

% max.strict_boundedE
tff(fact_7871_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% max.strict_boundedE
tff(fact_7872_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(B,C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_7873_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(ord_less_num(aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(ord_less_num(B),A)
         => ~ aa_num_o(ord_less_num(C),A) ) ) ).

% max.strict_boundedE
tff(fact_7874_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_7875_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_7876_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
    <=> ( ( A = ord_ma741700101516333627d_enat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7877_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7878_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = ord_max_rat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7879_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7880_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7881_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_7882_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_7883_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_7884_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_7885_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(ord_less_num(C),A)
     => aa_num_o(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_7886_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_7887_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_7888_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),B)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_7889_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_7890_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_7891_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(ord_less_num(C),B)
     => aa_num_o(ord_less_num(C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_7892_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_7893_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_7894_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
     => ( ! [Mi: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_7895_vebt__member_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $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,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_7896_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_7897_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xd != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_7898_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
        $ite(
          Xd = Mi3,
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $ite(
              Xd = Ma,
              one_one_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
                  one_one_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
                      one_one_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))),
                        $let(
                          h: nat,
                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_7899_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xd) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),Mi3,Xd),
        $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))),numeral_numeral_nat(bit0(one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xd = Mi3 )
                | ( Xd = Ma ) ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),Xd,Mi3)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(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,Va2))),numeral_numeral_nat(bit0(one))))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_7900_vebt__member_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => (Y) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2)
             => (Y) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => (Y) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $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,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_7901_vebt__member_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2)
           => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $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,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_7902_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != numeral_numeral_nat(bit0(one)) ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2)
             => ( Y != numeral_numeral_nat(bit0(one)) ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => ( Y != numeral_numeral_nat(bit0(one)) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                          $ite(
                            Xaa = Mi,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(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))),numeral_numeral_nat(bit0(one)))),
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),one_one_nat) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_7903_vebt__insert_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xd,Xaa) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B2)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B2))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) ) )
           => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(bit0(one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($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),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_7904_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_7905_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_7906_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd) ),
        one_one_nat,
        $ite(
          ( ( Xd = Mi3 )
          & ( Xd = Ma ) ),
          one_one_nat,
          $let(
            xn: nat,
            xn:= 
              $ite(Xd = Mi3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xd),
            $let(
              l: nat,
              l:= 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))),numeral_numeral_nat(bit0(one)))),
              $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))),numeral_numeral_nat(bit0(one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList,h)),l)),
                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),h),one_one_nat)),
                  one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_7907_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xd) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
        some_nat(Ma),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summary,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd),some_nat(Mi3),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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_7908_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xd) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
        some_nat(Mi3),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summary,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_7909_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xd) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd) ),
        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
        $ite(
          ( ( Xd = Mi3 )
          & ( Xd = Ma ) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
          $let(
            xn: nat,
            xn:= 
              $ite(Xd = Mi3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xd),
            $let(
              minn: nat,
              minn:= 
                $ite(Xd = Mi3,xn,Mi3),
              $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))),numeral_numeral_nat(bit0(one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(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,Va2))),numeral_numeral_nat(bit0(one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary,h),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_7910_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xd),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xd = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
             => ( ? [Va: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              one_one_nat,
                              $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList2,h)),l)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary2),h)),one_one_nat) ) ),
                                    one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_7911_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xd),Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xd = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            one_one_nat,
                            $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList2,h)),l)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary2),h)),one_one_nat) ) ),
                                  one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_7912_vebt__delete_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xd,Xaa) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B2)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B2: $o] : Xd = vEBT_Leaf((A4),(B2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf((A4),(B2)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) ) )
             => ( ! [Mi: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TrLst2),Smry2) ) )
               => ( ! [Mi: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma2 ) ),
                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(
                                                          xn = Ma2,
                                                          $let(
                                                            maxs: option_nat,
                                                            maxs:= vEBT_vebt_maxt(sn),
                                                            $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                          Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),sn) ),
                                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),Summary2) ) ) ),
                                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_7913_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma2 ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= 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))),numeral_numeral_nat(bit0(one)))),
                                    $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))),numeral_numeral_nat(bit0(one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary2),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_7914_vebt__pred_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xd,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xd = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ? [Va: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ( Y != $ite(
                        (B2),
                        some_nat(one_one_nat),
                        $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              some_nat(Ma2),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_7915_vebt__succ_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xd,Xaa) = Y )
     => ( ! [Uu2: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((Uu2),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B2),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_7916_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,aa_int_int(plus_plus_int(l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_7917_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,numeral_numeral_real(bit0(one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,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_7918_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),
          uminus1351360451143612070nteger(ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),ring_18347121197199848620nteger(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_7919_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(bit0(one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7920_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),
          uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(numeral_numeral_rat(bit0(one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,plus_plus_rat(l,one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7921_lowi__hT,axiom,
    ! [Xd: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(Xd,Nb),aa_nat_fun_nat_assn(aTP_Lamm_cd(Xd),Nb)),one_one_nat) ).

% lowi_hT
tff(fact_7922_highi__hT,axiom,
    ! [Xd: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_highi(Xd,Nb),aa_nat_fun_nat_assn(aTP_Lamm_ce(Xd),Nb)),one_one_nat) ).

% highi_hT
tff(fact_7923_div__half__nat,axiom,
    ! [Y: nat,Xd: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xd),Y)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xd),numeral_numeral_nat(bit0(one)))),Y)),
            $let(
              r: nat,
              r:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xd),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_7924_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2)
               => ( Y != numeral_numeral_nat(bit0(one)) ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),
                          $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))),numeral_numeral_nat(bit0(one)))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = Ma2 ) ) ),
                                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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),h),one_one_nat)),
                                one_one_nat ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_7925_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma2,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                    $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))),numeral_numeral_nat(bit0(one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),zero_zero_nat) ),
                                    zero_zero_nat ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_7926_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),Mi3,Xd),
          $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))),numeral_numeral_nat(bit0(one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
              & ~ ( ( Xd = Mi3 )
                  | ( Xd = Ma ) ) ),
              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(vEBT_T_i_n_s_e_r_t(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,Va2))),numeral_numeral_nat(bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,h)))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),one_one_nat)),
              one_one_nat ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_7927_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(bit0(one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),h),one_one_nat)),
                              one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_7928_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A),(B))),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_7929_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,T)),one_one_nat) ).

% minNull_bound
tff(fact_7930_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf((Uu),$true)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_7931_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($true,(Uv))) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_7932_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($false,$false)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_7933_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A),(B))),Xd) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_7934_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts),S)),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_7935_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb),Vc2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_7936_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_7937_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)),Xd) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_7938_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_7939_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary)),Xd) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_7940_insersimp_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),Y)),one_one_nat) ) ) ).

% insersimp'
tff(fact_7941_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_7942_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xd) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_7943_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% insert'_bound_height
tff(fact_7944_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xd) = Y )
     => ( ( ( Xd = vEBT_Leaf($false,$false) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv2: $o] : Xd = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] : Xd = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc)
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_7945_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xd: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc2)),Xd) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_7946_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(T),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% member_bound_height'
tff(fact_7947_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),Mi3,Xd),
        $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))),numeral_numeral_nat(bit0(one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xd = Mi3 )
                | ( Xd = Ma ) ) ),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(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,Va2))),numeral_numeral_nat(bit0(one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),one_one_nat)),
            one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_7948_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          Xd = Mi3,
          zero_zero_nat,
          $ite(
            Xd = Ma,
            zero_zero_nat,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
                zero_zero_nat,
                $ite(
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),Xd)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Ma) ),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_7949_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,K))) ).

% finite_Collect_less_nat
tff(fact_7950_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xd),Xaa) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    numeral_numeral_nat(aa_num_num(bit1,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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(aa_num_num(bit1,one))))),
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= 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))),numeral_numeral_nat(bit0(one)))),
                                          $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))),numeral_numeral_nat(bit0(one)))),
                                            $ite(
                                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                      $ite(
                                                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary2),h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary2,h),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                  $let(
                                                                    maxs: option_nat,
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                      $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                one_one_nat )) )),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                              one_one_nat ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_7951_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
        $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd) ),
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
            $ite(
              ( ( Xd = Mi3 )
              & ( Xd = Ma ) ),
              numeral_numeral_nat(aa_num_num(bit1,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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(aa_num_num(bit1,one))))),
                        $ite(Xd = Mi3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xd = Mi3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xd),
                  $let(
                    l: nat,
                    l:= 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))),numeral_numeral_nat(bit0(one)))),
                    $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))),numeral_numeral_nat(bit0(one)))),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(TreeList,h)),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l),
                            $let(
                              newlist: list_VEBT_VEBT,
                              newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                $ite(
                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary),h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary,h),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                        $ite(
                                          xn = Ma,
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                            $let(
                                              maxs: option_nat,
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                          one_one_nat )) )),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                        one_one_nat ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_7952_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xd),Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xd = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xd = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                              one_one_nat,
                              $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                                            $ite(
                                              ( ( maxlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= vEBT_vebt_succ(Summary2,h),
                                                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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary2),h))),one_one_nat)),
                                                  $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_7953_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_7954_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_7955_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_7956_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,T)),numeral_numeral_nat(aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_7957_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_7958_mint__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,T)),numeral_numeral_nat(aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_7959_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_7960_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi3: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_7961_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A7)
           => ( ! [B7: set_nat] :
                  ( aa_set_nat_o(ord_less_set_nat(B7),A7)
                 => aa_set_nat_o(P,B7) )
             => aa_set_nat_o(P,A7) ) )
       => aa_set_nat_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_7962_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(finite_finite_int,A7)
           => ( ! [B7: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B7),A7)
                 => aa_set_int_o(P,B7) )
             => aa_set_int_o(P,A7) ) )
       => aa_set_int_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_7963_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A7)
           => ( ! [B7: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B7),A7)
                 => aa_set_complex_o(P,B7) )
             => aa_set_complex_o(P,A7) ) )
       => aa_set_complex_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_7964_finite__psubset__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
           => ( ! [B7: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B7),A7)
                 => aa_set5643387332196575111eger_o(P,B7) )
             => aa_set5643387332196575111eger_o(P,A7) ) )
       => aa_set5643387332196575111eger_o(P,A2) ) ) ).

% finite_psubset_induct
tff(fact_7965_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xd) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_7966_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xd) = Y )
     => ( ! [A4: $o] :
            ( ? [B2: $o] : Xd = vEBT_Leaf((A4),(B2))
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A4),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_7967_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xd),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    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(plus_plus_nat,numeral_numeral_nat(bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),numeral_numeral_nat(aa_num_num(bit1,one)))),
                      $ite(
                        ( ( minlow != none_nat )
                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList,h)),l)),
                        $let(
                          pr: option_nat,
                          pr:= vEBT_vebt_pred(Summary,h),
                          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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),h))),one_one_nat)),
                            $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
                  one_one_nat )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_7968_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi3: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xd: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xd) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),Mi3),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
                    $let(
                      maxlow: option_nat,
                      maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                        $ite(
                          ( ( maxlow != none_nat )
                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList,h)),l)),
                          $let(
                            sc: option_nat,
                            sc:= vEBT_vebt_succ(Summary,h),
                            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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),h))),one_one_nat)),
                              $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
                  one_one_nat )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_7969_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xd),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xd = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ? [Va: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2)
                 => ( Y != one_one_nat ) )
               => ( ( ? [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                one_one_nat,
                                $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          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(plus_plus_nat,numeral_numeral_nat(bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,h)))),numeral_numeral_nat(aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= vEBT_vebt_pred(Summary2,h),
                                                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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary2),h))),one_one_nat)),
                                                  $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_7970_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                                      $ite(
                                        ( ( Xaa = Mi )
                                        & ( Xaa = Ma2 ) ),
                                        numeral_numeral_nat(aa_num_num(bit1,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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(aa_num_num(bit1,one))))),
                                                  $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= 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))),numeral_numeral_nat(bit0(one)))),
                                              $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))),numeral_numeral_nat(bit0(one)))),
                                                $ite(
                                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                          $ite(
                                                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary2),h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                      $let(
                                                                        maxs: option_nat,
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                          $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                    one_one_nat )) )),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                                                              $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_7971_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  one_one_nat,
                                  $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),
                                        $ite(
                                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,one))),
                                                $ite(
                                                  ( ( maxlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_s_u_c_c(nth_VEBT_VEBT(TreeList2,h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= vEBT_vebt_succ(Summary2,h),
                                                    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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary2),h))),one_one_nat)),
                                                      $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_7972_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    one_one_nat,
                                    $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),one_one_nat)),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                              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(plus_plus_nat,numeral_numeral_nat(bit0(one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,h)))),numeral_numeral_nat(aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(vEBT_T_p_r_e_d(nth_VEBT_VEBT(TreeList2,h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= vEBT_vebt_pred(Summary2,h),
                                                    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(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary2),h))),one_one_nat)),
                                                      $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_7973_pred__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_pred(T,Xd) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cg(T),Xd)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_7974_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_7975_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_7976_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_7977_succ__empty,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( ( vEBT_vebt_succ(T,Xd) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ch(T),Xd)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_7978_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_7979_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(ord_less_num(B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_7980_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7981_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_7982_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_7983_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_7984_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Q),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_7985_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(pure_assn((B))),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_7986_mod__h__bot__iff_I4_J,axiom,
    ! [Q2: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : ~ aa_Pro653082453696054616_nat_o(rep_assn(snga_assn_VEBT_VEBTi(Q2,Y)),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(4)
tff(fact_7987_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7988_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7989_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H4,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_7990_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_7991_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A2),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_7992_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_7993_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
            & ~ ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),S3)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7994_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),S3)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7995_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),S3)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7996_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( aa_set_num_o(finite_finite_num,S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),S3)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S3)
                  & aa_num_o(ord_less_num(Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7997_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7998_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),S3)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_7999_infinite__growing,axiom,
    ! [X7: set_Code_integer] :
      ( ( X7 != bot_bo3990330152332043303nteger )
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),X7)
           => ? [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),X7)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X7) ) ) ).

% infinite_growing
tff(fact_8000_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),X7)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X7)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X7) ) ) ).

% infinite_growing
tff(fact_8001_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X: rat] :
            ( aa_set_rat_o(member_rat(X),X7)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X7)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X7) ) ) ).

% infinite_growing
tff(fact_8002_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X: num] :
            ( aa_set_num_o(member_num(X),X7)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X7)
                & aa_num_o(ord_less_num(X),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X7) ) ) ).

% infinite_growing
tff(fact_8003_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X7)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X7)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X7) ) ) ).

% infinite_growing
tff(fact_8004_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X7)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X7)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X7) ) ) ).

% infinite_growing
tff(fact_8005_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_8006_vebt__succ_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xd,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B2),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_8007_vebt__pred_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xd,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A4),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = $ite(
                              (B2),
                              some_nat(one_one_nat),
                              $ite((A4),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  some_nat(Ma2),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_8008_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TreeList2),Summary2),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= 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))),numeral_numeral_nat(bit0(one)))),
                                        $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))),numeral_numeral_nat(bit0(one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary2),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_8009_vebt__delete_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xd,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B2)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,B2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,TrLst2),Smry2),Xaa)) ) )
                 => ( ! [Mi: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma2 ) ),
                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                            $ite(
                                                              xn = Ma2,
                                                              $let(
                                                                maxs: option_nat,
                                                                maxs:= vEBT_vebt_maxt(sn),
                                                                $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                              Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),sn) ),
                                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),Summary2) ) ) ),
                                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_8010_one__assn__raw_Ocases,axiom,
    ! [Xd: produc3658429121746597890et_nat] :
      ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] : Xd != produc7507926704131184380et_nat(H3,As) ).

% one_assn_raw.cases
tff(fact_8011_times__assn__raw_Ocases,axiom,
    ! [Xd: produc3112576180687729086et_nat] :
      ~ ! [P4: fun_Pr6460944699146269935_nat_o,Q7: fun_Pr6460944699146269935_nat_o,H3: heap_e7401611519738050253t_unit,As: set_nat] : Xd != produc7291855603632349880et_nat(P4,produc6017135503889078682et_nat(Q7,produc7507926704131184380et_nat(H3,As))) ).

% times_assn_raw.cases
tff(fact_8012_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc),Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                one_one_nat,
                                $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(nth_VEBT_VEBT(TreeList2,h)),l)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary2),h)),one_one_nat) ) ),
                                      one_one_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_8013_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) )
                   => ( ( Y = numeral_numeral_nat(bit0(one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one)))))),
                              $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))),numeral_numeral_nat(bit0(one)))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = Ma2 ) ) ),
                                    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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_8014_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xd = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A4: $o,B2: $o] :
                  ( ( Xd = vEBT_Leaf((A4),(B2)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vd2),Ve2),Xaa)) ) )
                 => ( ! [V3: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  one_one_nat,
                                  $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))),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(nth_VEBT_VEBT(TreeList2,h)),l)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary2),h)),one_one_nat) ) ),
                                        one_one_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_8015_vebt__insert_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xd,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B2)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B2))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(bit0(one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($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),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_8016_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts2),S2),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2),S2),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),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))),numeral_numeral_nat(bit0(one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_8017_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = numeral_numeral_nat(bit0(one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = numeral_numeral_nat(bit0(one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ( Y = numeral_numeral_nat(bit0(one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),
                              $ite(
                                Xaa = Mi,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                            one_one_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(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))),numeral_numeral_nat(bit0(one)))),
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))))),one_one_nat) )) )) )) )) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_8018_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xd),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                        $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))),numeral_numeral_nat(bit0(one)))),
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),zero_zero_nat) ),
                                        zero_zero_nat ) ) ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_8019_vebt__member_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xaa)) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2),Xaa)) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc),Xaa)) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $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,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_8020_vebt__member_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $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,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_8021_vebt__member_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xd),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2),Uz2),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $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,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_8022_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2),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)),numeral_numeral_nat(bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_8023_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2),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)),numeral_numeral_nat(bit0(one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_8024_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xd),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B2)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2) )
                 => ( ( (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)),numeral_numeral_nat(bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList2),S2),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_8025_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_8026_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_8027_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xd),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),zero_zero_nat,Va3),Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma2: nat,V3: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),aa_nat_nat(suc,V3),TreeList2),Vc),Xaa)) ) )
               => ~ ! [V3: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),numeral_numeral_nat(bit0(one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList2),Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_8028_monoseq__arctan__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_ci,Xd)) ) ).

% monoseq_arctan_series
tff(fact_8029_delete__correct,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xd)) = minus_minus_set_nat(vEBT_set_vebt(T),insert_nat(Xd,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_8030_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D)) ) ).

% foldr_zero
tff(fact_8031_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ).

% foldr_one
tff(fact_8032_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
           => ( X = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_8033_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,Nb: nat,Xd: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,Xd)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(T),insert_nat(Xd,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_8034_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Ys),I3)) )
       => ( 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,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_8035_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_cj,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_8036_foldr__length,axiom,
    ! [L: list_o] : aa_nat_nat(foldr_o_nat(aTP_Lamm_ck,L),zero_zero_nat) = size_size_list_o(L) ).

% foldr_length
tff(fact_8037_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_cl,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_8038_foldr__length,axiom,
    ! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_cm,L),zero_zero_nat) = size_size_list_int(L) ).

% foldr_length
tff(fact_8039_set__replicate,axiom,
    ! [Nb: nat,Xd: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xd)) = insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_8040_set__replicate,axiom,
    ! [Nb: nat,Xd: real] :
      ( ( Nb != zero_zero_nat )
     => ( set_real2(replicate_real(Nb,Xd)) = insert_real(Xd,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_8041_set__replicate,axiom,
    ! [Nb: nat,Xd: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,Xd)) = insert_nat(Xd,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_8042_set__replicate,axiom,
    ! [Nb: nat,Xd: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,Xd)) = insert_int(Xd,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_8043_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_8044_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_o,K: list_o,F: fun_o_fun_nat_nat,G: fun_o_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: nat,X: $o] :
              ( member_o((X),set_o2(L))
             => ( aa_nat_nat(aa_o_fun_nat_nat(F,(X)),A4) = aa_nat_nat(aa_o_fun_nat_nat(G,(X)),A4) ) )
         => ( aa_nat_nat(foldr_o_nat(F,L),A) = aa_nat_nat(foldr_o_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_8045_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_nat,K: list_nat,F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: nat,X: nat] :
              ( aa_set_nat_o(member_nat(X),set_nat2(L))
             => ( aa_nat_nat(aa_nat_fun_nat_nat(F,X),A4) = aa_nat_nat(aa_nat_fun_nat_nat(G,X),A4) ) )
         => ( aa_nat_nat(foldr_nat_nat(F,L),A) = aa_nat_nat(foldr_nat_nat(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_8046_foldr__cong,axiom,
    ! [A: real,B: real,L: list_real,K: list_real,F: fun_re7681423839302404990l_real,G: fun_re7681423839302404990l_real] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A4: real,X: real] :
              ( aa_set_real_o(member_real(X),set_real2(L))
             => ( aa_real_real(aa_rea7340605660300375871l_real(F,X),A4) = aa_real_real(aa_rea7340605660300375871l_real(G,X),A4) ) )
         => ( aa_real_real(foldr_real_real(F,L),A) = aa_real_real(foldr_real_real(G,K),B) ) ) ) ) ).

% foldr_cong
tff(fact_8047_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),X4) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B2,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8048_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B2: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),X4) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B2,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8049_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),X4) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B2,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8050_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B2: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A7)
                   => aa_num_o(ord_less_num(B2),X4) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B2,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8051_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),X4) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B2,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8052_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),X4) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B2,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8053_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),B2) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B2,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8054_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B2: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B2) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B2,A7)) ) ) )
         => aa_set_real_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8055_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A2)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),B2) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B2,A7)) ) ) )
         => aa_set_rat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8056_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A2)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B2: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A7)
                   => aa_num_o(ord_less_num(X4),B2) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B2,A7)) ) ) )
         => aa_set_num_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8057_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B2) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B2,A7)) ) ) )
         => aa_set_nat_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8058_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B2) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B2,A7)) ) ) )
         => aa_set_int_o(P,A2) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8059_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_8060_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_8061_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8062_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8063_remove__subset,axiom,
    ! [Xd: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),S3)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT))),S3) ) ).

% remove_subset
tff(fact_8064_remove__subset,axiom,
    ! [Xd: complex,S3: set_complex] :
      ( aa_set_complex_o(member_complex(Xd),S3)
     => aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(S3,insert_complex(Xd,bot_bot_set_complex))),S3) ) ).

% remove_subset
tff(fact_8065_remove__subset,axiom,
    ! [Xd: real,S3: set_real] :
      ( aa_set_real_o(member_real(Xd),S3)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S3,insert_real(Xd,bot_bot_set_real))),S3) ) ).

% remove_subset
tff(fact_8066_remove__subset,axiom,
    ! [Xd: int,S3: set_int] :
      ( aa_set_int_o(member_int(Xd),S3)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S3,insert_int(Xd,bot_bot_set_int))),S3) ) ).

% remove_subset
tff(fact_8067_remove__subset,axiom,
    ! [Xd: nat,S3: set_nat] :
      ( aa_set_nat_o(member_nat(Xd),S3)
     => aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(S3,insert_nat(Xd,bot_bot_set_nat))),S3) ) ).

% remove_subset
tff(fact_8068_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,Xd: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xd))),insert_real(Xd,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_8069_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,Xd: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xd))),insert_int(Xd,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_8070_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xd: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xd))),insert_VEBT_VEBT(Xd,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_8071_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xd: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xd)),insert_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_8072_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,Xd: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xd))),insert_nat(Xd,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_8073_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_cj,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_8074_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] : aa_nat_nat(foldr_o_nat(aTP_Lamm_ck,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_o(L)) ).

% foldr_length_aux
tff(fact_8075_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_cl,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_8076_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_cm,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_int(L)) ).

% foldr_length_aux
tff(fact_8077_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [T2: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T2),S3)
             => ( aa_set_VEBT_VEBT_o(P,T2)
               => ? [X4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(S3,T2))
                    & aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X4,T2)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8078_finite__induct__select,axiom,
    ! [S3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T2: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T2),S3)
             => ( aa_set_complex_o(P,T2)
               => ? [X4: complex] :
                    ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(S3,T2))
                    & aa_set_complex_o(P,insert_complex(X4,T2)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8079_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [T2: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T2),S3)
             => ( aa_set5643387332196575111eger_o(P,T2)
               => ? [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),minus_2355218937544613996nteger(S3,T2))
                    & aa_set5643387332196575111eger_o(P,insert_Code_integer(X4,T2)) ) ) )
         => aa_set5643387332196575111eger_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8080_finite__induct__select,axiom,
    ! [S3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T2: set_real] :
              ( aa_set_real_o(ord_less_set_real(T2),S3)
             => ( aa_set_real_o(P,T2)
               => ? [X4: real] :
                    ( aa_set_real_o(member_real(X4),minus_minus_set_real(S3,T2))
                    & aa_set_real_o(P,insert_real(X4,T2)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8081_finite__induct__select,axiom,
    ! [S3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T2: set_int] :
              ( aa_set_int_o(ord_less_set_int(T2),S3)
             => ( aa_set_int_o(P,T2)
               => ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),minus_minus_set_int(S3,T2))
                    & aa_set_int_o(P,insert_int(X4,T2)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8082_finite__induct__select,axiom,
    ! [S3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T2: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T2),S3)
             => ( aa_set_nat_o(P,T2)
               => ? [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(S3,T2))
                    & aa_set_nat_o(P,insert_nat(X4,T2)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8083_set__replicate__Suc,axiom,
    ! [Nb: nat,Xd: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xd)) = insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_8084_set__replicate__Suc,axiom,
    ! [Nb: nat,Xd: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xd)) = insert_real(Xd,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_8085_set__replicate__Suc,axiom,
    ! [Nb: nat,Xd: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xd)) = insert_nat(Xd,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_8086_set__replicate__Suc,axiom,
    ! [Nb: nat,Xd: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xd)) = insert_int(Xd,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_8087_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,Xd: vEBT_VEBT,B3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),insert_VEBT_VEBT(Xd,B3))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),B3),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A2),B3),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT))),B3),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B3)) ) ) ).

% psubset_insert_iff
tff(fact_8088_psubset__insert__iff,axiom,
    ! [A2: set_complex,Xd: complex,B3: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A2),insert_complex(Xd,B3))
    <=> $ite(
          aa_set_complex_o(member_complex(Xd),B3),
          aa_set_complex_o(ord_less_set_complex(A2),B3),
          $ite(aa_set_complex_o(member_complex(Xd),A2),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A2,insert_complex(Xd,bot_bot_set_complex))),B3),aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)) ) ) ).

% psubset_insert_iff
tff(fact_8089_psubset__insert__iff,axiom,
    ! [A2: set_real,Xd: real,B3: set_real] :
      ( aa_set_real_o(ord_less_set_real(A2),insert_real(Xd,B3))
    <=> $ite(
          aa_set_real_o(member_real(Xd),B3),
          aa_set_real_o(ord_less_set_real(A2),B3),
          $ite(aa_set_real_o(member_real(Xd),A2),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A2,insert_real(Xd,bot_bot_set_real))),B3),aa_set_real_o(ord_less_eq_set_real(A2),B3)) ) ) ).

% psubset_insert_iff
tff(fact_8090_psubset__insert__iff,axiom,
    ! [A2: set_int,Xd: int,B3: set_int] :
      ( aa_set_int_o(ord_less_set_int(A2),insert_int(Xd,B3))
    <=> $ite(
          aa_set_int_o(member_int(Xd),B3),
          aa_set_int_o(ord_less_set_int(A2),B3),
          $ite(aa_set_int_o(member_int(Xd),A2),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A2,insert_int(Xd,bot_bot_set_int))),B3),aa_set_int_o(ord_less_eq_set_int(A2),B3)) ) ) ).

% psubset_insert_iff
tff(fact_8091_psubset__insert__iff,axiom,
    ! [A2: set_nat,Xd: nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),insert_nat(Xd,B3))
    <=> $ite(
          aa_set_nat_o(member_nat(Xd),B3),
          aa_set_nat_o(ord_less_set_nat(A2),B3),
          $ite(aa_set_nat_o(member_nat(Xd),A2),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A2,insert_nat(Xd,bot_bot_set_nat))),B3),aa_set_nat_o(ord_less_eq_set_nat(A2),B3)) ) ) ).

% psubset_insert_iff
tff(fact_8092_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xd: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xd)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_8093_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xd: real] :
      set_real2(replicate_real(Nb,Xd)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real(Xd,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_8094_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xd: nat] :
      set_nat2(replicate_nat(Nb,Xd)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(Xd,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_8095_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xd: int] :
      set_int2(replicate_int(Nb,Xd)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(Xd,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_8096_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xd: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( insert_VEBT_VEBT(nth_VEBT_VEBT(L,I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Xd))) = insert_VEBT_VEBT(Xd,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8097_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xd: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Xd))) = insert_VEBT_VEBTi(Xd,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8098_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,Xd))) = insert_real(Xd,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8099_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,Xd: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( insert_o(aa_nat_o(nth_o(L),I),set_o2(list_update_o(L,I,(Xd)))) = insert_o((Xd),set_o2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8100_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,Xd))) = insert_nat(Xd,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8101_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,Xd: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,Xd))) = insert_int(Xd,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_8102_monoseq__realpow,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xd)) ) ) ).

% monoseq_realpow
tff(fact_8103_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_8104_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X: real,Y2: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ( aa_set_real_o(member_real(Y2),set_real2(Xs))
           => ( X = Y2 ) ) )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_real2(Xs))
           => ( X = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_8105_diff__preserves__multiset,axiom,
    ! [M7: fun_set_nat_nat,N2: fun_set_nat_nat] :
      ( finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_cn(M7)))
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun1332161875654541722_nat_o(aTP_Lamm_co(M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_8106_diff__preserves__multiset,axiom,
    ! [M7: fun_nat_nat,N2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_cp(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_cq(M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_8107_diff__preserves__multiset,axiom,
    ! [M7: fun_int_nat,N2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_cr(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_cs(M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_8108_diff__preserves__multiset,axiom,
    ! [M7: fun_complex_nat,N2: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ct(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_cu(M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_8109_diff__preserves__multiset,axiom,
    ! [M7: fun_Code_integer_nat,N2: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_cv(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_cw(M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_8110_add__mset__in__multiset,axiom,
    ! [M7: fun_set_nat_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_cn(M7)))
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_cx(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_8111_add__mset__in__multiset,axiom,
    ! [M7: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_cp(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cy(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_8112_add__mset__in__multiset,axiom,
    ! [M7: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_cr(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_cz(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_8113_add__mset__in__multiset,axiom,
    ! [M7: fun_complex_nat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ct(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_da(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_8114_add__mset__in__multiset,axiom,
    ! [M7: fun_Code_integer_nat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_cv(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_db(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_8115_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(bit0(one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_dc(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8116_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))))),aa_nat_nat(suc,Nb))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_dd,Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8117_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(bit0(one))))),aa_nat_nat(suc,Nb))) = groups6464643781859351333omplex(aTP_Lamm_de(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8118_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_8119_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8120_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,Nb: nat] :
      groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_assn,aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_assn(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8121_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] :
      groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8122_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,Nb: nat] :
      groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_Code_integer(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8123_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,M: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8124_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8125_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M),one_one_int,aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8126_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_df(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),A) ).

% mod_prod_eq
tff(fact_8127_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A2: set_nat] : aa_int_int(modulo_modulo_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_dg(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups705719431365010083at_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8128_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_dh(F),A),A2)),A) = aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(F,A2)),A) ).

% mod_prod_eq
tff(fact_8129_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_di(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8130_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_dj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8131_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8132_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: 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,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_dl(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8133_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_dm(G),Nb),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8134_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,M: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_dn(G),Nb),M),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),M)) ).

% prod.atLeastAtMost_rev
tff(fact_8135_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_assn,Nb: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_assn(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8136_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_8137_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_Code_integer,Nb: nat] : groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_Code_integer(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8138_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_complex,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8139_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_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,aa_set_nat_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_8140_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(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_8141_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = aa_assn_assn(times_times_assn(aa_nat_assn(G,aa_nat_nat(suc,Nb))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8142_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),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(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8143_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(G,aa_nat_nat(suc,Nb))),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8144_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,aa_nat_nat(suc,Nb))),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8145_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),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))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8146_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb))) = aa_int_int(times_times_int(aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8147_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_assn_assn(times_times_assn(aa_nat_assn(G,M)),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8148_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8149_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(G,M)),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8150_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M)),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8151_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8152_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_int_int(times_times_int(aa_nat_int(G,M)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8153_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_assn(G,aa_nat_nat(suc,Nb))) = aa_assn_assn(times_times_assn(aa_nat_assn(G,M)),groups6906906614972039071t_assn(aTP_Lamm_do(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8154_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),groups129246275422532515t_real(aTP_Lamm_dp(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8155_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_Code_integer(G,aa_nat_nat(suc,Nb))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(G,M)),groups3455450783089532116nteger(aTP_Lamm_dq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8156_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(G,M)),groups6464643781859351333omplex(aTP_Lamm_dr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8157_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),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,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_di(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8158_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(times_times_int(aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_dj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8159_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8160_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8161_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8162_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8163_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(plus_plus_int(one_one_int),Nb))
     => ( set_or1266510415728281911st_int(M,aa_int_int(plus_plus_int(one_one_int),Nb)) = insert_int(aa_int_int(plus_plus_int(one_one_int),Nb),set_or1266510415728281911st_int(M,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8164_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(plus_plus_int(I),one_one_int),J))) ).

% simp_from_to
tff(fact_8165_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_ds(F),A,B,one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8166_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_assn,A: nat,B: nat] : groups6906906614972039071t_assn(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1959793692361082170t_assn(aTP_Lamm_dt(F),A,B,one_one_assn) ).

% prod_atLeastAtMost_code
tff(fact_8167_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_du(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8168_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_Code_integer,A: nat,B: nat] : groups3455450783089532116nteger(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1084959871951514735nteger(aTP_Lamm_dv(F),A,B,one_one_Code_integer) ).

% prod_atLeastAtMost_code
tff(fact_8169_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_dw(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8170_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_dx(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8171_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_dy(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8172_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),groups1707563613775114915nt_nat(F,A2))
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_int_nat(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8173_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),groups861055069439313189ex_nat(F,A2))
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_complex_nat(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8174_even__prod__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),groups3190895334310489300er_nat(F,A2))
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_Code_integer_nat(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8175_even__prod__iff,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),groups858564598930262913ex_int(F,A2))
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_complex_int(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8176_even__prod__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),groups3188404863801439024er_int(F,A2))
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_Code_integer_int(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8177_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8178_even__prod__iff,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),groups705719431365010083at_int(F,A2))
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_int(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8179_even__prod__iff,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),groups1705073143266064639nt_int(F,A2))
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(F,X5)) ) ) ) ).

% even_prod_iff
tff(fact_8180_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_assn,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),groups6906906614972039071t_assn(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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8181_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_real,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8182_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_Code_integer,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),groups3455450783089532116nteger(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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8183_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_complex,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),groups6464643781859351333omplex(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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8184_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8185_prod_Oub__add__nat,axiom,
    ! [M: nat,Nb: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P3))) = aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),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),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8186_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8187_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_dz(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8188_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_ea,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8189_pochhammer__Suc__prod,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = groups6464643781859351333omplex(aTP_Lamm_eb(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8190_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)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ec,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8191_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_ed(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8192_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_ee(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8193_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_ef(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8194_pochhammer__prod__rev,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_eg(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8195_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_eh(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8196_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_ei(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8197_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_8198_prod_Oin__pairs,axiom,
    ! [G: fun_nat_assn,M: nat,Nb: nat] : groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = groups6906906614972039071t_assn(aTP_Lamm_ej(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8199_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_ek(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8200_prod_Oin__pairs,axiom,
    ! [G: fun_nat_Code_integer,M: nat,Nb: nat] : groups3455450783089532116nteger(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = groups3455450783089532116nteger(aTP_Lamm_el(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8201_prod_Oin__pairs,axiom,
    ! [G: fun_nat_complex,M: nat,Nb: nat] : groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = groups6464643781859351333omplex(aTP_Lamm_em(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8202_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_en(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8203_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_eo(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ).

% prod.in_pairs
tff(fact_8204_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_ee(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8205_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_ef(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8206_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_eg(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8207_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)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_eh(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8208_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_ei(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8209_filter__preserves__multiset,axiom,
    ! [M7: fun_set_nat_nat,P: fun_set_nat_o] :
      ( finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_cn(M7)))
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ep(M7),P))) ) ).

% filter_preserves_multiset
tff(fact_8210_filter__preserves__multiset,axiom,
    ! [M7: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_cp(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_eq(M7),P))) ) ).

% filter_preserves_multiset
tff(fact_8211_filter__preserves__multiset,axiom,
    ! [M7: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_cr(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_er(M7),P))) ) ).

% filter_preserves_multiset
tff(fact_8212_filter__preserves__multiset,axiom,
    ! [M7: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ct(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_es(M7),P))) ) ).

% filter_preserves_multiset
tff(fact_8213_filter__preserves__multiset,axiom,
    ! [M7: fun_Code_integer_nat,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_cv(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_et(M7),P))) ) ).

% filter_preserves_multiset
tff(fact_8214_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),
        aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ).

% and_int.simps
tff(fact_8215_and__int_Oelims,axiom,
    ! [Xd: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xd),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xd),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xd)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xaa) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xd)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xd),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% and_int.elims
tff(fact_8216_prod_Oinsert,axiom,
    ! [A2: set_real,Xd: real,G: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xd),A2)
       => ( groups1155561341820557179l_assn(G,insert_real(Xd,A2)) = aa_assn_assn(times_times_assn(aa_real_assn(G,Xd)),groups1155561341820557179l_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8217_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xd: vEBT_VEBT,G: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2)
       => ( groups569574905686396791T_assn(G,insert_VEBT_VEBT(Xd,A2)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(G,Xd)),groups569574905686396791T_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8218_prod_Oinsert,axiom,
    ! [A2: set_nat,Xd: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xd),A2)
       => ( groups6906906614972039071t_assn(G,insert_nat(Xd,A2)) = aa_assn_assn(times_times_assn(aa_nat_assn(G,Xd)),groups6906906614972039071t_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8219_prod_Oinsert,axiom,
    ! [A2: set_int,Xd: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xd),A2)
       => ( groups7882442080178216443t_assn(G,insert_int(Xd,A2)) = aa_assn_assn(times_times_assn(aa_int_assn(G,Xd)),groups7882442080178216443t_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8220_prod_Oinsert,axiom,
    ! [A2: set_complex,Xd: complex,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ~ aa_set_complex_o(member_complex(Xd),A2)
       => ( groups4150731942483176573x_assn(G,insert_complex(Xd,A2)) = aa_assn_assn(times_times_assn(aa_complex_assn(G,Xd)),groups4150731942483176573x_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8221_prod_Oinsert,axiom,
    ! [A2: set_Code_integer,Xd: code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xd),A2)
       => ( groups1304777262505850412r_assn(G,insert_Code_integer(Xd,A2)) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,Xd)),groups1304777262505850412r_assn(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8222_prod_Oinsert,axiom,
    ! [A2: set_real,Xd: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ~ aa_set_real_o(member_real(Xd),A2)
       => ( groups1681761925125756287l_real(G,insert_real(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xd)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8223_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,Xd: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2)
       => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xd)),groups2703838992350267259T_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8224_prod_Oinsert,axiom,
    ! [A2: set_nat,Xd: nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Xd),A2)
       => ( groups129246275422532515t_real(G,insert_nat(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xd)),groups129246275422532515t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8225_prod_Oinsert,axiom,
    ! [A2: set_int,Xd: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ~ aa_set_int_o(member_int(Xd),A2)
       => ( groups2316167850115554303t_real(G,insert_int(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xd)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod.insert
tff(fact_8226_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_eu(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8227_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_ev(A),B),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_VEBT_VEBT_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8228_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_ew(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8229_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_ex(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8230_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_ey(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8231_prod_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_ez(A),B),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S3),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8232_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fa(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8233_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_fb(A),B),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_VEBT_VEBT_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8234_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8235_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fd(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_8236_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_fe(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8237_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_ff(A),B),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_VEBT_VEBT_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8238_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_fg(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8239_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_fh(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8240_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_fi(A),B),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_complex_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8241_prod_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_fj(A),B),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S3),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8242_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fk(A),B),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8243_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aTP_Lamm_fl(A),B),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_VEBT_VEBT_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8244_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fm(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8245_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fn(A),B),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_8246_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A)),groups4696554848551431203al_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8247_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A)),groups6361806394783013919BT_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8248_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A)),groups1707563613775114915nt_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8249_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A)),groups861055069439313189ex_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8250_dvd__prodI,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Code_integer_nat(F,A)),groups3190895334310489300er_nat(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8251_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A)),groups4694064378042380927al_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8252_dvd__prodI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_VEBT_VEBT_int(F,A)),groups6359315924273963643BT_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8253_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A)),groups858564598930262913ex_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8254_dvd__prodI,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Code_integer_int(F,A)),groups3188404863801439024er_int(F,A2)) ) ) ).

% dvd_prodI
tff(fact_8255_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A)),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ) ).

% dvd_prodI
tff(fact_8256_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fo),A2) = one_one_nat ).

% prod.neutral_const
tff(fact_8257_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_fp,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8258_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_fq,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8259_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups6464643781859351333omplex(F,A2) = zero_zero_complex )
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
            & ( aa_nat_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8260_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups7440179247065528705omplex(F,A2) = zero_zero_complex )
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
            & ( aa_int_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8261_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups3708469109370488835omplex(F,A2) = zero_zero_complex )
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
            & ( aa_complex_complex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8262_prod__zero__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Co1900716531650368264omplex] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( groups862514429393162674omplex(F,A2) = zero_zero_complex )
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
            & ( aa_Cod5621350732091595593omplex(F,X5) = zero_zero_complex ) ) ) ) ).

% prod_zero_iff
tff(fact_8263_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups129246275422532515t_real(F,A2) = zero_zero_real )
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
            & ( aa_nat_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8264_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
            & ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8265_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
            & ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8266_prod__zero__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( groups9004974159866482096r_real(F,A2) = zero_zero_real )
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
            & ( aa_Code_integer_real(F,X5) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_8267_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( groups73079841787564623at_rat(F,A2) = zero_zero_rat )
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
            & ( aa_nat_rat(F,X5) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8268_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat )
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
            & ( aa_int_rat(F,X5) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_8269_prod_Oempty,axiom,
    ! [G: fun_real_assn] : groups1155561341820557179l_assn(G,bot_bot_set_real) = one_one_assn ).

% prod.empty
tff(fact_8270_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8271_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8272_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_8273_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_8274_prod_Oempty,axiom,
    ! [G: fun_nat_assn] : groups6906906614972039071t_assn(G,bot_bot_set_nat) = one_one_assn ).

% prod.empty
tff(fact_8275_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8276_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8277_prod_Oempty,axiom,
    ! [G: fun_int_assn] : groups7882442080178216443t_assn(G,bot_bot_set_int) = one_one_assn ).

% prod.empty
tff(fact_8278_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8279_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_assn] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_8280_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_assn] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_8281_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_assn] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups4150731942483176573x_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_8282_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1304777262505850412r_assn(G,A2) = one_one_assn ) ) ).

% prod.infinite
tff(fact_8283_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8284_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8285_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8286_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_8287_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8288_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_8289_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( ( B = aa_real_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups4696554848551431203al_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8290_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: nat,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
       => ( ( B = aa_VEBT_VEBT_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups6361806394783013919BT_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8291_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(A),A2)
       => ( ( B = aa_int_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups1707563613775114915nt_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8292_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: nat,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => ( ( B = aa_complex_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups861055069439313189ex_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8293_dvd__prod__eqI,axiom,
    ! [A2: set_Code_integer,A: code_integer,B: nat,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => ( ( B = aa_Code_integer_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),groups3190895334310489300er_nat(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8294_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(A),A2)
       => ( ( B = aa_real_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups4694064378042380927al_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8295_dvd__prod__eqI,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B: int,F: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2)
       => ( ( B = aa_VEBT_VEBT_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups6359315924273963643BT_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8296_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: int,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(A),A2)
       => ( ( B = aa_complex_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups858564598930262913ex_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8297_dvd__prod__eqI,axiom,
    ! [A2: set_Code_integer,A: code_integer,B: int,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A2)
       => ( ( B = aa_Code_integer_int(F,A) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),groups3188404863801439024er_int(F,A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8298_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ( ( B = aa_nat_nat(F,A) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ) ) ).

% dvd_prod_eqI
tff(fact_8299_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( groups1707563613775114915nt_nat(F,A2) = one_one_nat )
      <=> ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
           => ( aa_int_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8300_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( groups861055069439313189ex_nat(F,A2) = one_one_nat )
      <=> ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
           => ( aa_complex_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8301_prod__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( groups3190895334310489300er_nat(F,A2) = one_one_nat )
      <=> ! [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
           => ( aa_Code_integer_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8302_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A2) = one_one_nat )
      <=> ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
           => ( aa_nat_nat(F,X5) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8303_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A2))
      <=> ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8304_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A2))
      <=> ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8305_prod__pos__nat__iff,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups3190895334310489300er_nat(F,A2))
      <=> ! [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8306_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))
      <=> ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8307_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_aq,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_8308_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ( aa_nat_nat(G,X) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8309_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ( aa_nat_int(G,X) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8310_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ( aa_int_int(G,X) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_8311_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_assn,A2: set_nat] :
      ( ( groups6906906614972039071t_assn(G,A2) != one_one_assn )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8312_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_assn,A2: set_real] :
      ( ( groups1155561341820557179l_assn(G,A2) != one_one_assn )
     => ~ ! [A4: real] :
            ( aa_set_real_o(member_real(A4),A2)
           => ( aa_real_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8313_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_assn,A2: set_int] :
      ( ( groups7882442080178216443t_assn(G,A2) != one_one_assn )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8314_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_assn,A2: set_VEBT_VEBT] :
      ( ( groups569574905686396791T_assn(G,A2) != one_one_assn )
     => ~ ! [A4: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
           => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8315_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_assn,A2: set_complex] :
      ( ( groups4150731942483176573x_assn(G,A2) != one_one_assn )
     => ~ ! [A4: complex] :
            ( aa_set_complex_o(member_complex(A4),A2)
           => ( aa_complex_assn(G,A4) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8316_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A2: set_nat] :
      ( ( groups129246275422532515t_real(G,A2) != one_one_real )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A2)
           => ( aa_nat_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8317_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups1681761925125756287l_real(G,A2) != one_one_real )
     => ~ ! [A4: real] :
            ( aa_set_real_o(member_real(A4),A2)
           => ( aa_real_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8318_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups2316167850115554303t_real(G,A2) != one_one_real )
     => ~ ! [A4: int] :
            ( aa_set_int_o(member_int(A4),A2)
           => ( aa_int_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8319_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A2: set_VEBT_VEBT] :
      ( ( groups2703838992350267259T_real(G,A2) != one_one_real )
     => ~ ! [A4: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
           => ( aa_VEBT_VEBT_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8320_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups766887009212190081x_real(G,A2) != one_one_real )
     => ~ ! [A4: complex] :
            ( aa_set_complex_o(member_complex(A4),A2)
           => ( aa_complex_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8321_prod__dvd__prod,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [A4: real] :
          ( aa_set_real_o(member_real(A4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A4)),aa_real_nat(G,A4)) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8322_prod__dvd__prod,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [A4: int] :
          ( aa_set_int_o(member_int(A4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A4)),aa_int_nat(G,A4)) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8323_prod__dvd__prod,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [A4: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A4)),aa_VEBT_VEBT_nat(G,A4)) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8324_prod__dvd__prod,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [A4: complex] :
          ( aa_set_complex_o(member_complex(A4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A4)),aa_complex_nat(G,A4)) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8325_prod__dvd__prod,axiom,
    ! [A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [A4: real] :
          ( aa_set_real_o(member_real(A4),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A4)),aa_real_int(G,A4)) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8326_prod__dvd__prod,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [A4: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_VEBT_VEBT_int(F,A4)),aa_VEBT_VEBT_int(G,A4)) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8327_prod__dvd__prod,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [A4: complex] :
          ( aa_set_complex_o(member_complex(A4),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A4)),aa_complex_int(G,A4)) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8328_prod__dvd__prod,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [A4: nat] :
          ( aa_set_nat_o(member_nat(A4),A2)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A4)),aa_nat_nat(G,A4)) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),A2)) ) ).

% prod_dvd_prod
tff(fact_8329_prod__dvd__prod,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [A4: nat] :
          ( aa_set_nat_o(member_nat(A4),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(F,A4)),aa_nat_int(G,A4)) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8330_prod__dvd__prod,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [A4: int] :
          ( aa_set_int_o(member_int(A4),A2)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(F,A4)),aa_int_int(G,A4)) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(G,A2)) ) ).

% prod_dvd_prod
tff(fact_8331_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A2)),aa_set_nat_nat(groups708209901874060359at_nat(H),A2)) ).

% prod.distrib
tff(fact_8332_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A2: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_fs(G),H),A2) = aa_int_int(times_times_int(groups705719431365010083at_int(G,A2)),groups705719431365010083at_int(H,A2)) ).

% prod.distrib
tff(fact_8333_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(G),H),A2) = aa_int_int(times_times_int(groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(H,A2)) ).

% prod.distrib
tff(fact_8334_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fu(F),Nb)),A2) ).

% prod_power_distrib
tff(fact_8335_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A2: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A2)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_fv(F),Nb),A2) ).

% prod_power_distrib
tff(fact_8336_prod__power__distrib,axiom,
    ! [F: fun_int_int,A2: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A2)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_fw(F),Nb),A2) ).

% prod_power_distrib
tff(fact_8337_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_nonneg
tff(fact_8338_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_nonneg
tff(fact_8339_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_nonneg
tff(fact_8340_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A2)
         => ( 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,A2)),groups129246275422532515t_real(G,A2)) ) ).

% prod_mono
tff(fact_8341_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A2)
         => ( 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,A2)),groups1681761925125756287l_real(G,A2)) ) ).

% prod_mono
tff(fact_8342_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A2)
         => ( 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,A2)),groups2316167850115554303t_real(G,A2)) ) ).

% prod_mono
tff(fact_8343_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2)) ) ).

% prod_mono
tff(fact_8344_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),A2)
         => ( 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,A2)),groups766887009212190081x_real(G,A2)) ) ).

% prod_mono
tff(fact_8345_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A2)
         => ( 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,A2)),groups73079841787564623at_rat(G,A2)) ) ).

% prod_mono
tff(fact_8346_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( aa_set_real_o(member_real(I3),A2)
         => ( 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,A2)),groups4061424788464935467al_rat(G,A2)) ) ).

% prod_mono
tff(fact_8347_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A2)
         => ( 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,A2)),groups1072433553688619179nt_rat(G,A2)) ) ).

% prod_mono
tff(fact_8348_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2)) ) ).

% prod_mono
tff(fact_8349_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( aa_set_complex_o(member_complex(I3),A2)
         => ( 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,A2)),groups225925009352817453ex_rat(G,A2)) ) ).

% prod_mono
tff(fact_8350_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) ) ).

% prod_pos
tff(fact_8351_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2)) ) ).

% prod_pos
tff(fact_8352_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2)) ) ).

% prod_pos
tff(fact_8353_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8354_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8355_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8356_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2703838992350267259T_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8357_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A2)) ) ).

% prod_ge_1
tff(fact_8358_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8359_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8360_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8361_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups5726676334696518183BT_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8362_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A2)) ) ).

% prod_ge_1
tff(fact_8363_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_complex(F,X4) = zero_zero_complex ) )
       => ( groups6464643781859351333omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8364_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_complex(F,X4) = zero_zero_complex ) )
       => ( groups7440179247065528705omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8365_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_complex(F,X4) = zero_zero_complex ) )
       => ( groups3708469109370488835omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8366_prod__zero,axiom,
    ! [A2: set_Code_integer,F: fun_Co1900716531650368264omplex] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ? [X4: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A2)
            & ( aa_Cod5621350732091595593omplex(F,X4) = zero_zero_complex ) )
       => ( groups862514429393162674omplex(F,A2) = zero_zero_complex ) ) ) ).

% prod_zero
tff(fact_8367_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_real(F,X4) = zero_zero_real ) )
       => ( groups129246275422532515t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8368_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_real(F,X4) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8369_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A2)
            & ( aa_complex_real(F,X4) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8370_prod__zero,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ? [X4: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A2)
            & ( aa_Code_integer_real(F,X4) = zero_zero_real ) )
       => ( groups9004974159866482096r_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8371_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A2)
            & ( aa_nat_rat(F,X4) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8372_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A2)
            & ( aa_int_rat(F,X4) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8373_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xd: fun_real_complex,Y: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_fx(I6),Xd)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_fx(I6),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fy(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8374_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,Xd: fun_VE5569070535936029429omplex,Y: fun_VE5569070535936029429omplex] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun677759971260770784VEBT_o(aTP_Lamm_fz(I6),Xd)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun677759971260770784VEBT_o(aTP_Lamm_fz(I6),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun677759971260770784VEBT_o(aa_fun5749144123862529590VEBT_o(aTP_Lamm_ga(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8375_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xd: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_gb(I6),Xd)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_gb(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_gc(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8376_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xd: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_gd(I6),Xd)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_gd(I6),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ge(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8377_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xd: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_gf(I6),Xd)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_gf(I6),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_gg(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8378_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Code_integer,Xd: fun_Co1900716531650368264omplex,Y: fun_Co1900716531650368264omplex] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun144181944155979022eger_o(aTP_Lamm_gh(I6),Xd)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun144181944155979022eger_o(aTP_Lamm_gh(I6),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun144181944155979022eger_o(aa_fun9138836748425859551eger_o(aTP_Lamm_gi(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8379_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xd: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_gj(I6),Xd)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_gj(I6),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_gk(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8380_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,Xd: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_gl(I6),Xd)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_gl(I6),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_gm(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8381_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xd: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_gn(I6),Xd)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_gn(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_go(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8382_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xd: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_gp(I6),Xd)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_gp(I6),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_gq(I6),Xd),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8383_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xd: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_gr(I6),Xd)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_gr(I6),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_gs(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8384_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,Xd: fun_VEBT_VEBT_rat,Y: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_gt(I6),Xd)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_gt(I6),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_gu(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8385_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xd: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_gv(I6),Xd)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_gv(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_gw(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8386_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xd: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_gx(I6),Xd)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_gx(I6),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_gy(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8387_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xd: fun_complex_rat,Y: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_gz(I6),Xd)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_gz(I6),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_ha(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8388_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_Code_integer,Xd: fun_Code_integer_rat,Y: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_hb(I6),Xd)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_hb(I6),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_hc(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8389_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xd: fun_real_assn,Y: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aTP_Lamm_hd(I6),Xd)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aTP_Lamm_hd(I6),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aa_fun4959323011771448382real_o(aTP_Lamm_he(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8390_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_VEBT_VEBT,Xd: fun_VEBT_VEBT_assn,Y: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aTP_Lamm_hf(I6),Xd)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aTP_Lamm_hf(I6),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aa_fun876682331325704374VEBT_o(aTP_Lamm_hg(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8391_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xd: fun_nat_assn,Y: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_hh(I6),Xd)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_hh(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_hi(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8392_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xd: fun_int_assn,Y: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_hj(I6),Xd)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_hj(I6),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_hk(I6),Xd),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8393_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_assn,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1155561341820557179l_assn(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_hl(A2),P))) = groups1155561341820557179l_assn(aa_fun3688174147639456858l_assn(aTP_Lamm_hm(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8394_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups569574905686396791T_assn(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hn(A2),P))) = groups569574905686396791T_assn(aa_fun8737571391773076682T_assn(aTP_Lamm_ho(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8395_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_assn,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_hp(A2),P))) = groups6906906614972039071t_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_hq(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8396_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_assn,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_hr(A2),P))) = groups7882442080178216443t_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_hs(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8397_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_assn,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups4150731942483176573x_assn(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ht(A2),P))) = groups4150731942483176573x_assn(aa_fun6270938053774919130x_assn(aTP_Lamm_hu(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8398_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: fun_Co4490034186337852034r_assn,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1304777262505850412r_assn(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_hv(A2),P))) = groups1304777262505850412r_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_hw(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8399_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_hl(A2),P))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_hx(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8400_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2703838992350267259T_real(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hn(A2),P))) = groups2703838992350267259T_real(aa_fun3597767889293385422T_real(aTP_Lamm_hy(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8401_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_real,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_hp(A2),P))) = groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_hz(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8402_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_hr(A2),P))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_ia(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_8403_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8404_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8405_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8406_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8407_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A2)),one_one_real) ) ).

% prod_le_1
tff(fact_8408_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8409_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8410_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8411_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8412_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),one_one_rat) ) ).

% prod_le_1
tff(fact_8413_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X)),aa_nat_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8414_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X)),aa_int_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H,S3)),groups1072433553688619179nt_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8415_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X)),aa_complex_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S3)),groups225925009352817453ex_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8416_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X22),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_Code_integer_rat(H,X)),aa_Code_integer_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2555765274223993564er_rat(H,S3)),groups2555765274223993564er_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8417_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S3: set_nat,H: fun_nat_assn,G: fun_nat_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X1: assn,Y1: assn,X22: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X1),X22)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y1),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X1),Y1)),aa_assn_assn(times_times_assn(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_nat_assn(H,X)),aa_nat_assn(G,X)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups6906906614972039071t_assn(H,S3)),groups6906906614972039071t_assn(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8418_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S3: set_int,H: fun_int_assn,G: fun_int_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X1: assn,Y1: assn,X22: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X1),X22)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y1),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X1),Y1)),aa_assn_assn(times_times_assn(X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_int_assn(H,X)),aa_int_assn(G,X)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups7882442080178216443t_assn(H,S3)),groups7882442080178216443t_assn(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8419_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S3: set_complex,H: fun_complex_assn,G: fun_complex_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X1: assn,Y1: assn,X22: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X1),X22)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y1),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X1),Y1)),aa_assn_assn(times_times_assn(X22),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_complex_assn(H,X)),aa_complex_assn(G,X)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups4150731942483176573x_assn(H,S3)),groups4150731942483176573x_assn(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8420_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S3: set_Code_integer,H: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X1: assn,Y1: assn,X22: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X1),X22)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y1),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X1),Y1)),aa_assn_assn(times_times_assn(X22),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_Code_integer_assn(H,X)),aa_Code_integer_assn(G,X)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,groups1304777262505850412r_assn(H,S3)),groups1304777262505850412r_assn(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8421_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X1: real,Y1: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X)),aa_nat_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups129246275422532515t_real(H,S3)),groups129246275422532515t_real(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8422_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X1: real,Y1: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X)),aa_int_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups2316167850115554303t_real(H,S3)),groups2316167850115554303t_real(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_8423_prod_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_assn,Xd: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1155561341820557179l_assn(G,insert_real(Xd,A2)) = $ite(aa_set_real_o(member_real(Xd),A2),groups1155561341820557179l_assn(G,A2),aa_assn_assn(times_times_assn(aa_real_assn(G,Xd)),groups1155561341820557179l_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8424_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,Xd: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups569574905686396791T_assn(G,insert_VEBT_VEBT(Xd,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2),groups569574905686396791T_assn(G,A2),aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(G,Xd)),groups569574905686396791T_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8425_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_assn,Xd: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,insert_nat(Xd,A2)) = $ite(aa_set_nat_o(member_nat(Xd),A2),groups6906906614972039071t_assn(G,A2),aa_assn_assn(times_times_assn(aa_nat_assn(G,Xd)),groups6906906614972039071t_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8426_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_assn,Xd: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,insert_int(Xd,A2)) = $ite(aa_set_int_o(member_int(Xd),A2),groups7882442080178216443t_assn(G,A2),aa_assn_assn(times_times_assn(aa_int_assn(G,Xd)),groups7882442080178216443t_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8427_prod_Oinsert__if,axiom,
    ! [A2: set_complex,G: fun_complex_assn,Xd: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups4150731942483176573x_assn(G,insert_complex(Xd,A2)) = $ite(aa_set_complex_o(member_complex(Xd),A2),groups4150731942483176573x_assn(G,A2),aa_assn_assn(times_times_assn(aa_complex_assn(G,Xd)),groups4150731942483176573x_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8428_prod_Oinsert__if,axiom,
    ! [A2: set_Code_integer,G: fun_Co4490034186337852034r_assn,Xd: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1304777262505850412r_assn(G,insert_Code_integer(Xd,A2)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Xd),A2),groups1304777262505850412r_assn(G,A2),aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,Xd)),groups1304777262505850412r_assn(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8429_prod_Oinsert__if,axiom,
    ! [A2: set_real,G: fun_real_real,Xd: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1681761925125756287l_real(G,insert_real(Xd,A2)) = $ite(aa_set_real_o(member_real(Xd),A2),groups1681761925125756287l_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xd)),groups1681761925125756287l_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8430_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xd: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xd,A2)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2),groups2703838992350267259T_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xd)),groups2703838992350267259T_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8431_prod_Oinsert__if,axiom,
    ! [A2: set_nat,G: fun_nat_real,Xd: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups129246275422532515t_real(G,insert_nat(Xd,A2)) = $ite(aa_set_nat_o(member_nat(Xd),A2),groups129246275422532515t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Xd)),groups129246275422532515t_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8432_prod_Oinsert__if,axiom,
    ! [A2: set_int,G: fun_int_real,Xd: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,insert_int(Xd,A2)) = $ite(aa_set_int_o(member_int(Xd),A2),groups2316167850115554303t_real(G,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,Xd)),groups2316167850115554303t_real(G,A2))) ) ) ).

% prod.insert_if
tff(fact_8433_prod__dvd__prod__subset2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,B3)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A4)),aa_real_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8434_prod__dvd__prod__subset2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_VEBT_VEBT_nat(F,A4)),aa_VEBT_VEBT_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups6361806394783013919BT_nat(F,A2)),groups6361806394783013919BT_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8435_prod__dvd__prod__subset2,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B3)
       => ( ! [A4: int] :
              ( aa_set_int_o(member_int(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A4)),aa_int_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8436_prod__dvd__prod__subset2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => ( ! [A4: complex] :
              ( aa_set_complex_o(member_complex(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A4)),aa_complex_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8437_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => ( ! [A4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Code_integer_nat(F,A4)),aa_Code_integer_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A2)),groups3190895334310489300er_nat(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8438_prod__dvd__prod__subset2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_int,G: fun_real_int] :
      ( aa_set_real_o(finite_finite_real,B3)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A4)),aa_real_int(G,A4)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8439_prod__dvd__prod__subset2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_VEBT_VEBT_int(F,A4)),aa_VEBT_VEBT_int(G,A4)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups6359315924273963643BT_int(F,A2)),groups6359315924273963643BT_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8440_prod__dvd__prod__subset2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => ( ! [A4: complex] :
              ( aa_set_complex_o(member_complex(A4),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A4)),aa_complex_int(G,A4)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8441_prod__dvd__prod__subset2,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => ( ! [A4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A2)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Code_integer_int(F,A4)),aa_Code_integer_int(G,A4)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A2)),groups3188404863801439024er_int(G,B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8442_prod__dvd__prod__subset2,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => ( ! [A4: nat] :
              ( aa_set_nat_o(member_nat(A4),A2)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A4)),aa_nat_nat(G,A4)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(G),B3)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8443_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A2)),groups1707563613775114915nt_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8444_prod__dvd__prod__subset,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8445_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(F,A2)),groups3190895334310489300er_nat(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8446_prod__dvd__prod__subset,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8447_prod__dvd__prod__subset,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(F,A2)),groups3188404863801439024er_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8448_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)),aa_set_nat_nat(groups708209901874060359at_nat(F),B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8449_prod__dvd__prod__subset,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8450_prod__dvd__prod__subset,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(F,B3)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8451_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_real_o(finite_finite_real,T3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
               => aa_set_real_o(member_real(aa_real_real(J,A4)),minus_minus_set_real(T4,T3)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T3))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T3))
                   => aa_set_real_o(member_real(aa_real_real(I,B2)),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T3)
                       => ( aa_real_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S3)
                         => ( aa_real_assn(H,aa_real_real(J,A4)) = aa_real_assn(G,A4) ) )
                     => ( groups1155561341820557179l_assn(G,S3) = groups1155561341820557179l_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8452_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_VEBT_VEBT,S3: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T4: set_VEBT_VEBT,G: fun_real_assn,H: fun_VEBT_VEBT_assn] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T4,T3)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T3))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T3))
                   => aa_set_real_o(member_real(aa_VEBT_VEBT_real(I,B2)),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T3)
                       => ( aa_VEBT_VEBT_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S3)
                         => ( aa_VEBT_VEBT_assn(H,aa_real_VEBT_VEBT(J,A4)) = aa_real_assn(G,A4) ) )
                     => ( groups1155561341820557179l_assn(G,S3) = groups569574905686396791T_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8453_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_real,S3: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T4: set_real,G: fun_VEBT_VEBT_assn,H: fun_real_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
     => ( aa_set_real_o(finite_finite_real,T3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
               => aa_set_real_o(member_real(aa_VEBT_VEBT_real(J,A4)),minus_minus_set_real(T4,T3)) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T3))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( aa_set_real_o(member_real(B2),minus_minus_set_real(T4,T3))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S3,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S4)
                     => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: real] :
                        ( aa_set_real_o(member_real(B2),T3)
                       => ( aa_real_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S3)
                         => ( aa_real_assn(H,aa_VEBT_VEBT_real(J,A4)) = aa_VEBT_VEBT_assn(G,A4) ) )
                     => ( groups569574905686396791T_assn(G,S3) = groups1155561341820557179l_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8454_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T4: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A4)),minus_5127226145743854075T_VEBT(T4,T3)) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T3))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(T4,T3))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S3,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S4)
                     => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),T3)
                       => ( aa_VEBT_VEBT_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S3)
                         => ( aa_VEBT_VEBT_assn(H,aa_VEB5832953974749077948T_VEBT(J,A4)) = aa_VEBT_VEBT_assn(G,A4) ) )
                     => ( groups569574905686396791T_assn(G,S3) = groups569574905686396791T_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8455_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_assn,H: fun_int_assn] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
               => aa_set_int_o(member_int(aa_real_int(J,A4)),minus_minus_set_int(T4,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T3))
                 => ( aa_real_int(J,aa_int_real(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T3))
                   => aa_set_real_o(member_real(aa_int_real(I,B2)),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S3)
                         => ( aa_int_assn(H,aa_real_int(J,A4)) = aa_real_assn(G,A4) ) )
                     => ( groups1155561341820557179l_assn(G,S3) = groups7882442080178216443t_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8456_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_int,S3: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T4: set_int,G: fun_VEBT_VEBT_assn,H: fun_int_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
               => aa_set_int_o(member_int(aa_VEBT_VEBT_int(J,A4)),minus_minus_set_int(T4,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T3))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T4,T3))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S3,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S4)
                     => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S3)
                         => ( aa_int_assn(H,aa_VEBT_VEBT_int(J,A4)) = aa_VEBT_VEBT_assn(G,A4) ) )
                     => ( groups569574905686396791T_assn(G,S3) = groups7882442080178216443t_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8457_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_assn,H: fun_complex_assn] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T4,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T3))
                 => ( aa_real_complex(J,aa_complex_real(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T3))
                   => aa_set_real_o(member_real(aa_complex_real(I,B2)),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S3)
                         => ( aa_complex_assn(H,aa_real_complex(J,A4)) = aa_real_assn(G,A4) ) )
                     => ( groups1155561341820557179l_assn(G,S3) = groups4150731942483176573x_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8458_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_complex,S3: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T4: set_complex,G: fun_VEBT_VEBT_assn,H: fun_complex_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
               => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(J,A4)),minus_811609699411566653omplex(T4,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T3))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T4,T3))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S3,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S4)
                     => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S3)
                         => ( aa_complex_assn(H,aa_VEBT_VEBT_complex(J,A4)) = aa_VEBT_VEBT_assn(G,A4) ) )
                     => ( groups569574905686396791T_assn(G,S3) = groups4150731942483176573x_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8459_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_Code_integer,S3: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T4: set_Code_integer,G: fun_real_assn,H: fun_Co4490034186337852034r_assn] :
      ( aa_set_real_o(finite_finite_real,S4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T3)
       => ( ! [A4: real] :
              ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A4)) = A4 ) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(S3,S4))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_real_Code_integer(J,A4)),minus_2355218937544613996nteger(T4,T3)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T3))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T3))
                   => aa_set_real_o(member_real(aa_Code_integer_real(I,B2)),minus_minus_set_real(S3,S4)) )
               => ( ! [A4: real] :
                      ( aa_set_real_o(member_real(A4),S4)
                     => ( aa_real_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T3)
                       => ( aa_Code_integer_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: real] :
                          ( aa_set_real_o(member_real(A4),S3)
                         => ( aa_Code_integer_assn(H,aa_real_Code_integer(J,A4)) = aa_real_assn(G,A4) ) )
                     => ( groups1155561341820557179l_assn(G,S3) = groups1304777262505850412r_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8460_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T3: set_Code_integer,S3: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T4: set_Code_integer,G: fun_VEBT_VEBT_assn,H: fun_Co4490034186337852034r_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T3)
       => ( ! [A4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A4)) = A4 ) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(S3,S4))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_VEB5935388676523804373nteger(J,A4)),minus_2355218937544613996nteger(T4,T3)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T3))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T4,T3))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B2)),minus_5127226145743854075T_VEBT(S3,S4)) )
               => ( ! [A4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S4)
                     => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T3)
                       => ( aa_Code_integer_assn(H,B2) = one_one_assn ) )
                   => ( ! [A4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),S3)
                         => ( aa_Code_integer_assn(H,aa_VEB5935388676523804373nteger(J,A4)) = aa_VEBT_VEBT_assn(G,A4) ) )
                     => ( groups569574905686396791T_assn(G,S3) = groups1304777262505850412r_assn(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8461_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ib(G)))) = groups7882442080178216443t_assn(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8462_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups4150731942483176573x_assn(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_ic(G)))) = groups4150731942483176573x_assn(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8463_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_id(G)))) = groups1304777262505850412r_assn(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8464_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ie(G)))) = groups2316167850115554303t_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8465_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_if(G)))) = groups766887009212190081x_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8466_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_ig(G)))) = groups9004974159866482096r_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8467_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ih(G)))) = groups1072433553688619179nt_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8468_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_ii(G)))) = groups225925009352817453ex_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8469_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A2,collect_Code_integer(aTP_Lamm_ij(G)))) = groups2555765274223993564er_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8470_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ik(G)))) = groups1707563613775114915nt_nat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8471_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( aa_set_real_o(member_real(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8472_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I6)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I))
         => ( ! [I3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8473_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8474_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8475_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( aa_set_complex_o(member_complex(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
         => ( ! [I3: complex] :
                ( aa_set_complex_o(member_complex(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8476_less__1__prod2,axiom,
    ! [I6: set_Code_integer,I: code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I6)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_Code_integer_real(F,I))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_Code_integer_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups9004974159866482096r_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8477_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( aa_set_real_o(member_real(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
         => ( ! [I3: real] :
                ( aa_set_real_o(member_real(I3),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8478_less__1__prod2,axiom,
    ! [I6: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I6)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I))
         => ( ! [I3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8479_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8480_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( aa_set_int_o(member_int(I),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_8481_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I6)
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_VEBT_VEBT_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2703838992350267259T_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8482_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8483_less__1__prod,axiom,
    ! [I6: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I6)
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_Code_integer_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups9004974159866482096r_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8484_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8485_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups129246275422532515t_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8486_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8487_less__1__prod,axiom,
    ! [I6: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I6)
     => ( ( I6 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_VEBT_VEBT_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups5726676334696518183BT_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8488_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( aa_set_complex_o(member_complex(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8489_less__1__prod,axiom,
    ! [I6: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I6)
     => ( ( I6 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8490_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( aa_set_real_o(member_real(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_8491_prod_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B3),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C5,B3))
                 => ( aa_real_assn(H,B2) = one_one_assn ) )
             => ( ( groups1155561341820557179l_assn(G,A2) = groups1155561341820557179l_assn(H,B3) )
              <=> ( groups1155561341820557179l_assn(G,C5) = groups1155561341820557179l_assn(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8492_prod_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C5)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B3),C5)
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C5,B3))
                 => ( aa_VEBT_VEBT_assn(H,B2) = one_one_assn ) )
             => ( ( groups569574905686396791T_assn(G,A2) = groups569574905686396791T_assn(H,B3) )
              <=> ( groups569574905686396791T_assn(G,C5) = groups569574905686396791T_assn(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8493_prod_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: fun_int_assn,H: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B3),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B3))
                 => ( aa_int_assn(H,B2) = one_one_assn ) )
             => ( ( groups7882442080178216443t_assn(G,A2) = groups7882442080178216443t_assn(H,B3) )
              <=> ( groups7882442080178216443t_assn(G,C5) = groups7882442080178216443t_assn(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8494_prod_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C5,B3))
                 => ( aa_complex_assn(H,B2) = one_one_assn ) )
             => ( ( groups4150731942483176573x_assn(G,A2) = groups4150731942483176573x_assn(H,B3) )
              <=> ( groups4150731942483176573x_assn(G,C5) = groups4150731942483176573x_assn(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8495_prod_Osame__carrier,axiom,
    ! [C5: set_Code_integer,A2: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C5)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C5,A2))
               => ( aa_Code_integer_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B3))
                 => ( aa_Code_integer_assn(H,B2) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,A2) = groups1304777262505850412r_assn(H,B3) )
              <=> ( groups1304777262505850412r_assn(G,C5) = groups1304777262505850412r_assn(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8496_prod_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B3),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_real(G,A4) = one_one_real ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C5,B3))
                 => ( aa_real_real(H,B2) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B3) )
              <=> ( groups1681761925125756287l_real(G,C5) = groups1681761925125756287l_real(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8497_prod_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C5)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B3),C5)
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_real(G,A4) = one_one_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C5,B3))
                 => ( aa_VEBT_VEBT_real(H,B2) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,A2) = groups2703838992350267259T_real(H,B3) )
              <=> ( groups2703838992350267259T_real(G,C5) = groups2703838992350267259T_real(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8498_prod_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B3),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_real(G,A4) = one_one_real ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B3))
                 => ( aa_int_real(H,B2) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,A2) = groups2316167850115554303t_real(H,B3) )
              <=> ( groups2316167850115554303t_real(G,C5) = groups2316167850115554303t_real(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8499_prod_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = one_one_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C5,B3))
                 => ( aa_complex_real(H,B2) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B3) )
              <=> ( groups766887009212190081x_real(G,C5) = groups766887009212190081x_real(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8500_prod_Osame__carrier,axiom,
    ! [C5: set_Code_integer,A2: set_Code_integer,B3: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C5)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C5,A2))
               => ( aa_Code_integer_real(G,A4) = one_one_real ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B3))
                 => ( aa_Code_integer_real(H,B2) = one_one_real ) )
             => ( ( groups9004974159866482096r_real(G,A2) = groups9004974159866482096r_real(H,B3) )
              <=> ( groups9004974159866482096r_real(G,C5) = groups9004974159866482096r_real(H,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8501_prod_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B3),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C5,B3))
                 => ( aa_real_assn(H,B2) = one_one_assn ) )
             => ( ( groups1155561341820557179l_assn(G,C5) = groups1155561341820557179l_assn(H,C5) )
               => ( groups1155561341820557179l_assn(G,A2) = groups1155561341820557179l_assn(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8502_prod_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C5)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B3),C5)
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C5,B3))
                 => ( aa_VEBT_VEBT_assn(H,B2) = one_one_assn ) )
             => ( ( groups569574905686396791T_assn(G,C5) = groups569574905686396791T_assn(H,C5) )
               => ( groups569574905686396791T_assn(G,A2) = groups569574905686396791T_assn(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8503_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: fun_int_assn,H: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B3),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B3))
                 => ( aa_int_assn(H,B2) = one_one_assn ) )
             => ( ( groups7882442080178216443t_assn(G,C5) = groups7882442080178216443t_assn(H,C5) )
               => ( groups7882442080178216443t_assn(G,A2) = groups7882442080178216443t_assn(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8504_prod_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C5,B3))
                 => ( aa_complex_assn(H,B2) = one_one_assn ) )
             => ( ( groups4150731942483176573x_assn(G,C5) = groups4150731942483176573x_assn(H,C5) )
               => ( groups4150731942483176573x_assn(G,A2) = groups4150731942483176573x_assn(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8505_prod_Osame__carrierI,axiom,
    ! [C5: set_Code_integer,A2: set_Code_integer,B3: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C5)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C5,A2))
               => ( aa_Code_integer_assn(G,A4) = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B3))
                 => ( aa_Code_integer_assn(H,B2) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(G,C5) = groups1304777262505850412r_assn(H,C5) )
               => ( groups1304777262505850412r_assn(G,A2) = groups1304777262505850412r_assn(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8506_prod_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C5)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),C5)
       => ( aa_set_real_o(ord_less_eq_set_real(B3),C5)
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),minus_minus_set_real(C5,A2))
               => ( aa_real_real(G,A4) = one_one_real ) )
           => ( ! [B2: real] :
                  ( aa_set_real_o(member_real(B2),minus_minus_set_real(C5,B3))
                 => ( aa_real_real(H,B2) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C5) = groups1681761925125756287l_real(H,C5) )
               => ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8507_prod_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),C5)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B3),C5)
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),minus_5127226145743854075T_VEBT(C5,A2))
               => ( aa_VEBT_VEBT_real(G,A4) = one_one_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(C5,B3))
                 => ( aa_VEBT_VEBT_real(H,B2) = one_one_real ) )
             => ( ( groups2703838992350267259T_real(G,C5) = groups2703838992350267259T_real(H,C5) )
               => ( groups2703838992350267259T_real(G,A2) = groups2703838992350267259T_real(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8508_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),C5)
       => ( aa_set_int_o(ord_less_eq_set_int(B3),C5)
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),minus_minus_set_int(C5,A2))
               => ( aa_int_real(G,A4) = one_one_real ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B3))
                 => ( aa_int_real(H,B2) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,C5) = groups2316167850115554303t_real(H,C5) )
               => ( groups2316167850115554303t_real(G,A2) = groups2316167850115554303t_real(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8509_prod_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),C5)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B3),C5)
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),minus_811609699411566653omplex(C5,A2))
               => ( aa_complex_real(G,A4) = one_one_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C5,B3))
                 => ( aa_complex_real(H,B2) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C5) = groups766887009212190081x_real(H,C5) )
               => ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8510_prod_Osame__carrierI,axiom,
    ! [C5: set_Code_integer,A2: set_Code_integer,B3: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),C5)
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),minus_2355218937544613996nteger(C5,A2))
               => ( aa_Code_integer_real(G,A4) = one_one_real ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B3))
                 => ( aa_Code_integer_real(H,B2) = one_one_real ) )
             => ( ( groups9004974159866482096r_real(G,C5) = groups9004974159866482096r_real(H,C5) )
               => ( groups9004974159866482096r_real(G,A2) = groups9004974159866482096r_real(H,B3) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8511_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_assn(G,X) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,S3) = groups7882442080178216443t_assn(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8512_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_assn(G,X) = one_one_assn ) )
         => ( groups4150731942483176573x_assn(G,S3) = groups4150731942483176573x_assn(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8513_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_assn(G,X) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,S3) = groups1304777262505850412r_assn(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8514_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_real(G,X) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8515_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8516_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_real(G,X) = one_one_real ) )
         => ( groups9004974159866482096r_real(G,S3) = groups9004974159866482096r_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8517_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_rat(G,X) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8518_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_rat(G,X) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8519_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_rat(G,X) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,S3) = groups2555765274223993564er_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8520_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_nat(G,X) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8521_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_assn(G,X) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(G,T4) = groups7882442080178216443t_assn(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8522_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_assn(G,X) = one_one_assn ) )
         => ( groups4150731942483176573x_assn(G,T4) = groups4150731942483176573x_assn(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8523_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_assn(G,X) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(G,T4) = groups1304777262505850412r_assn(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8524_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_real(G,X) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8525_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8526_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_real(G,X) = one_one_real ) )
         => ( groups9004974159866482096r_real(G,T4) = groups9004974159866482096r_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8527_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_rat(G,X) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,T4) = groups1072433553688619179nt_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8528_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_rat(G,X) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T4) = groups225925009352817453ex_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8529_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_rat(G,X) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,T4) = groups2555765274223993564er_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8530_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_nat(G,X) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T4) = groups1707563613775114915nt_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8531_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_assn,G: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T4,S3))
             => ( aa_real_assn(H,X) = one_one_assn ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S3)
               => ( aa_real_assn(G,X) = aa_real_assn(H,X) ) )
           => ( groups1155561341820557179l_assn(G,S3) = groups1155561341820557179l_assn(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8532_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_assn,G: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T4,S3))
             => ( aa_VEBT_VEBT_assn(H,X) = one_one_assn ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S3)
               => ( aa_VEBT_VEBT_assn(G,X) = aa_VEBT_VEBT_assn(H,X) ) )
           => ( groups569574905686396791T_assn(G,S3) = groups569574905686396791T_assn(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8533_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_assn(H,X) = one_one_assn ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ( aa_int_assn(G,X) = aa_int_assn(H,X) ) )
           => ( groups7882442080178216443t_assn(G,S3) = groups7882442080178216443t_assn(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8534_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_assn,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_assn(H,X) = one_one_assn ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ( aa_complex_assn(G,X) = aa_complex_assn(H,X) ) )
           => ( groups4150731942483176573x_assn(G,S3) = groups4150731942483176573x_assn(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8535_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_assn(H,X) = one_one_assn ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => ( aa_Code_integer_assn(G,X) = aa_Code_integer_assn(H,X) ) )
           => ( groups1304777262505850412r_assn(G,S3) = groups1304777262505850412r_assn(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8536_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T4,S3))
             => ( aa_real_real(H,X) = one_one_real ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S3)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups1681761925125756287l_real(G,S3) = groups1681761925125756287l_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8537_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T4,S3))
             => ( aa_VEBT_VEBT_real(H,X) = one_one_real ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S3)
               => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
           => ( groups2703838992350267259T_real(G,S3) = groups2703838992350267259T_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8538_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_real(H,X) = one_one_real ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ( aa_int_real(G,X) = aa_int_real(H,X) ) )
           => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8539_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_real(H,X) = one_one_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8540_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_real(H,X) = one_one_real ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => ( aa_Code_integer_real(G,X) = aa_Code_integer_real(H,X) ) )
           => ( groups9004974159866482096r_real(G,S3) = groups9004974159866482096r_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8541_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_assn,H: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T4,S3))
             => ( aa_real_assn(G,X) = one_one_assn ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S3)
               => ( aa_real_assn(G,X) = aa_real_assn(H,X) ) )
           => ( groups1155561341820557179l_assn(G,T4) = groups1155561341820557179l_assn(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8542_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,H: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T4,S3))
             => ( aa_VEBT_VEBT_assn(G,X) = one_one_assn ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S3)
               => ( aa_VEBT_VEBT_assn(G,X) = aa_VEBT_VEBT_assn(H,X) ) )
           => ( groups569574905686396791T_assn(G,T4) = groups569574905686396791T_assn(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8543_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_assn,H: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_assn(G,X) = one_one_assn ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ( aa_int_assn(G,X) = aa_int_assn(H,X) ) )
           => ( groups7882442080178216443t_assn(G,T4) = groups7882442080178216443t_assn(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8544_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_assn,H: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_assn(G,X) = one_one_assn ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ( aa_complex_assn(G,X) = aa_complex_assn(H,X) ) )
           => ( groups4150731942483176573x_assn(G,T4) = groups4150731942483176573x_assn(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8545_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co4490034186337852034r_assn,H: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_assn(G,X) = one_one_assn ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => ( aa_Code_integer_assn(G,X) = aa_Code_integer_assn(H,X) ) )
           => ( groups1304777262505850412r_assn(G,T4) = groups1304777262505850412r_assn(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8546_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T4)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T4,S3))
             => ( aa_real_real(G,X) = one_one_real ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S3)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups1681761925125756287l_real(G,T4) = groups1681761925125756287l_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8547_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T4,S3))
             => ( aa_VEBT_VEBT_real(G,X) = one_one_real ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S3)
               => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H,X) ) )
           => ( groups2703838992350267259T_real(G,T4) = groups2703838992350267259T_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8548_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T4)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T4,S3))
             => ( aa_int_real(G,X) = one_one_real ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ( aa_int_real(G,X) = aa_int_real(H,X) ) )
           => ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8549_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T4,S3))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8550_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T4,S3))
             => ( aa_Code_integer_real(G,X) = one_one_real ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S3)
               => ( aa_Code_integer_real(G,X) = aa_Code_integer_real(H,X) ) )
           => ( groups9004974159866482096r_real(G,T4) = groups9004974159866482096r_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8551_prod_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_assn] :
      ( aa_set_int_o(ord_less_eq_set_int(B3),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups7882442080178216443t_assn(G,A2) = aa_assn_assn(times_times_assn(groups7882442080178216443t_assn(G,minus_minus_set_int(A2,B3))),groups7882442080178216443t_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8552_prod_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_assn] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B3),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups4150731942483176573x_assn(G,A2) = aa_assn_assn(times_times_assn(groups4150731942483176573x_assn(G,minus_811609699411566653omplex(A2,B3))),groups4150731942483176573x_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8553_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups1304777262505850412r_assn(G,A2) = aa_assn_assn(times_times_assn(groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A2,B3))),groups1304777262505850412r_assn(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8554_prod_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B3),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups2316167850115554303t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A2,B3))),groups2316167850115554303t_real(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8555_prod_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B3),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,B3))),groups766887009212190081x_real(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8556_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups9004974159866482096r_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A2,B3))),groups9004974159866482096r_real(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8557_prod_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B3),A2)
     => ( aa_set_int_o(finite_finite_int,A2)
       => ( groups1707563613775114915nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A2,B3))),groups1707563613775114915nt_nat(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8558_prod_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B3),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups861055069439313189ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A2,B3))),groups861055069439313189ex_nat(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8559_prod_Osubset__diff,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B3),A2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
       => ( groups3190895334310489300er_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(G,minus_2355218937544613996nteger(A2,B3))),groups3190895334310489300er_nat(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8560_prod_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B3),A2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A2)
       => ( groups858564598930262913ex_int(G,A2) = aa_int_int(times_times_int(groups858564598930262913ex_int(G,minus_811609699411566653omplex(A2,B3))),groups858564598930262913ex_int(G,B3)) ) ) ) ).

% prod.subset_diff
tff(fact_8561_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_VEBT_VEBT_real(F,I3)),aa_VEBT_VEBT_real(G,I3)) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8562_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),A2)
           => ( 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_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
       => ( ( A2 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8563_prod__mono__strict,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I3))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,I3)),aa_Code_integer_real(G,I3)) ) )
       => ( ( A2 != bot_bo3990330152332043303nteger )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups9004974159866482096r_real(F,A2)),groups9004974159866482096r_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8564_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),A2)
           => ( 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_real,aa_real_real(F,I3)),aa_real_real(G,I3)) ) )
       => ( ( A2 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8565_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A2)
           => ( 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_real,aa_nat_real(F,I3)),aa_nat_real(G,I3)) ) )
       => ( ( A2 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups129246275422532515t_real(F,A2)),groups129246275422532515t_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8566_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A2)
           => ( 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_real,aa_int_real(F,I3)),aa_int_real(G,I3)) ) )
       => ( ( A2 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8567_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ! [I3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I3),A2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_VEBT_VEBT_rat(F,I3)),aa_VEBT_VEBT_rat(G,I3)) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8568_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ! [I3: complex] :
            ( aa_set_complex_o(member_complex(I3),A2)
           => ( 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_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
       => ( ( A2 != bot_bot_set_complex )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8569_prod__mono__strict,axiom,
    ! [A2: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,I3)),aa_Code_integer_rat(G,I3)) ) )
       => ( ( A2 != bot_bo3990330152332043303nteger )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2555765274223993564er_rat(F,A2)),groups2555765274223993564er_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8570_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ! [I3: real] :
            ( aa_set_real_o(member_real(I3),A2)
           => ( 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_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
       => ( ( A2 != bot_bot_set_real )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).

% prod_mono_strict
tff(fact_8571_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xd: vEBT_VEBT,G: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2)
       => ( groups569574905686396791T_assn(G,A2) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(G,Xd)),groups569574905686396791T_assn(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_8572_prod_Oremove,axiom,
    ! [A2: set_complex,Xd: complex,G: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xd),A2)
       => ( groups4150731942483176573x_assn(G,A2) = aa_assn_assn(times_times_assn(aa_complex_assn(G,Xd)),groups4150731942483176573x_assn(G,minus_811609699411566653omplex(A2,insert_complex(Xd,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_8573_prod_Oremove,axiom,
    ! [A2: set_Code_integer,Xd: code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xd),A2)
       => ( groups1304777262505850412r_assn(G,A2) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,Xd)),groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xd,bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_8574_prod_Oremove,axiom,
    ! [A2: set_real,Xd: real,G: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xd),A2)
       => ( groups1155561341820557179l_assn(G,A2) = aa_assn_assn(times_times_assn(aa_real_assn(G,Xd)),groups1155561341820557179l_assn(G,minus_minus_set_real(A2,insert_real(Xd,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_8575_prod_Oremove,axiom,
    ! [A2: set_int,Xd: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( aa_set_int_o(member_int(Xd),A2)
       => ( groups7882442080178216443t_assn(G,A2) = aa_assn_assn(times_times_assn(aa_int_assn(G,Xd)),groups7882442080178216443t_assn(G,minus_minus_set_int(A2,insert_int(Xd,bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_8576_prod_Oremove,axiom,
    ! [A2: set_nat,Xd: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(Xd),A2)
       => ( groups6906906614972039071t_assn(G,A2) = aa_assn_assn(times_times_assn(aa_nat_assn(G,Xd)),groups6906906614972039071t_assn(G,minus_minus_set_nat(A2,insert_nat(Xd,bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_8577_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,Xd: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xd),A2)
       => ( groups2703838992350267259T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xd)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% prod.remove
tff(fact_8578_prod_Oremove,axiom,
    ! [A2: set_complex,Xd: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( aa_set_complex_o(member_complex(Xd),A2)
       => ( groups766887009212190081x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xd)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xd,bot_bot_set_complex)))) ) ) ) ).

% prod.remove
tff(fact_8579_prod_Oremove,axiom,
    ! [A2: set_Code_integer,Xd: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xd),A2)
       => ( groups9004974159866482096r_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,Xd)),groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xd,bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_8580_prod_Oremove,axiom,
    ! [A2: set_real,Xd: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( aa_set_real_o(member_real(Xd),A2)
       => ( groups1681761925125756287l_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xd)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(Xd,bot_bot_set_real)))) ) ) ) ).

% prod.remove
tff(fact_8581_prod_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_assn,Xd: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups569574905686396791T_assn(G,insert_VEBT_VEBT(Xd,A2)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(G,Xd)),groups569574905686396791T_assn(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.insert_remove
tff(fact_8582_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_assn,Xd: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups4150731942483176573x_assn(G,insert_complex(Xd,A2)) = aa_assn_assn(times_times_assn(aa_complex_assn(G,Xd)),groups4150731942483176573x_assn(G,minus_811609699411566653omplex(A2,insert_complex(Xd,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_8583_prod_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: fun_Co4490034186337852034r_assn,Xd: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups1304777262505850412r_assn(G,insert_Code_integer(Xd,A2)) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,Xd)),groups1304777262505850412r_assn(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xd,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_8584_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_assn,Xd: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1155561341820557179l_assn(G,insert_real(Xd,A2)) = aa_assn_assn(times_times_assn(aa_real_assn(G,Xd)),groups1155561341820557179l_assn(G,minus_minus_set_real(A2,insert_real(Xd,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_8585_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: fun_int_assn,Xd: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( groups7882442080178216443t_assn(G,insert_int(Xd,A2)) = aa_assn_assn(times_times_assn(aa_int_assn(G,Xd)),groups7882442080178216443t_assn(G,minus_minus_set_int(A2,insert_int(Xd,bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_8586_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: fun_nat_assn,Xd: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups6906906614972039071t_assn(G,insert_nat(Xd,A2)) = aa_assn_assn(times_times_assn(aa_nat_assn(G,Xd)),groups6906906614972039071t_assn(G,minus_minus_set_nat(A2,insert_nat(Xd,bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_8587_prod_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xd: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,Xd)),groups2703838992350267259T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(Xd,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.insert_remove
tff(fact_8588_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: fun_complex_real,Xd: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( groups766887009212190081x_real(G,insert_complex(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,Xd)),groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,insert_complex(Xd,bot_bot_set_complex)))) ) ) ).

% prod.insert_remove
tff(fact_8589_prod_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: fun_Co3960896333048504454r_real,Xd: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( groups9004974159866482096r_real(G,insert_Code_integer(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(G,Xd)),groups9004974159866482096r_real(G,minus_2355218937544613996nteger(A2,insert_Code_integer(Xd,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_8590_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: fun_real_real,Xd: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( groups1681761925125756287l_real(G,insert_real(Xd,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,Xd)),groups1681761925125756287l_real(G,minus_minus_set_real(A2,insert_real(Xd,bot_bot_set_real)))) ) ) ).

% prod.insert_remove
tff(fact_8591_prod_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_assn,C: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups569574905686396791T_assn(aa_fun7127563710338807932T_assn(aa_fun7778866309842214326T_assn(aTP_Lamm_il(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(B,A)),groups569574905686396791T_assn(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups569574905686396791T_assn(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.delta_remove
tff(fact_8592_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_assn,C: fun_complex_assn] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups4150731942483176573x_assn(aa_fun8818621774317227492x_assn(aa_fun9098329248302834228x_assn(aTP_Lamm_im(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_assn_assn(times_times_assn(aa_complex_assn(B,A)),groups4150731942483176573x_assn(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups4150731942483176573x_assn(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_8593_prod_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn,C: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_in(A),B),C),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S3),aa_assn_assn(times_times_assn(aa_Code_integer_assn(B,A)),groups1304777262505850412r_assn(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups1304777262505850412r_assn(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.delta_remove
tff(fact_8594_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_assn,C: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1155561341820557179l_assn(aa_fun2649397157800988384l_assn(aa_fun1876152720653945902l_assn(aTP_Lamm_io(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_assn_assn(times_times_assn(aa_real_assn(B,A)),groups1155561341820557179l_assn(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1155561341820557179l_assn(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_8595_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: fun_int_assn,C: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ip(A),B),C),S3) = $ite(aa_set_int_o(member_int(A),S3),aa_assn_assn(times_times_assn(aa_int_assn(B,A)),groups7882442080178216443t_assn(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))),groups7882442080178216443t_assn(C,minus_minus_set_int(S3,insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_8596_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_assn,C: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_iq(A),B),C),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_assn_assn(times_times_assn(aa_nat_assn(B,A)),groups6906906614972039071t_assn(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))),groups6906906614972039071t_assn(C,minus_minus_set_nat(S3,insert_nat(A,bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_8597_prod_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2703838992350267259T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ir(A),B),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(B,A)),groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))),groups2703838992350267259T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% prod.delta_remove
tff(fact_8598_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_is(A),B),C),S3) = $ite(aa_set_complex_o(member_complex(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(B,A)),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))),groups766887009212190081x_real(C,minus_811609699411566653omplex(S3,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% prod.delta_remove
tff(fact_8599_prod_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real,C: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups9004974159866482096r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_it(A),B),C),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Code_integer_real(B,A)),groups9004974159866482096r_real(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups9004974159866482096r_real(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.delta_remove
tff(fact_8600_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: fun_real_real,C: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_iu(A),B),C),S3) = $ite(aa_set_real_o(member_real(A),S3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(B,A)),groups1681761925125756287l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))),groups1681761925125756287l_real(C,minus_minus_set_real(S3,insert_real(A,bot_bot_set_real)))) ) ) ).

% prod.delta_remove
tff(fact_8601_prod__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B3)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B3)
       => ( ! [B2: real] :
              ( aa_set_real_o(member_real(B2),minus_minus_set_real(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B2)) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A4)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8602_prod__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B3)
       => ( ! [B2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_VEBT_VEBT_real(F,B2)) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,A4)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2703838992350267259T_real(F,A2)),groups2703838992350267259T_real(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8603_prod__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B3)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B2)) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A4)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8604_prod__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B2)) )
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A4)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8605_prod__mono2,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_Code_integer_real(F,B2)) )
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,A4)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups9004974159866482096r_real(F,A2)),groups9004974159866482096r_real(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8606_prod__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B3)
     => ( aa_set_real_o(ord_less_eq_set_real(A2),B3)
       => ( ! [B2: real] :
              ( aa_set_real_o(member_real(B2),minus_minus_set_real(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B2)) )
         => ( ! [A4: real] :
                ( aa_set_real_o(member_real(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8607_prod__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A2),B3)
       => ( ! [B2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_VEBT_VEBT_rat(F,B2)) )
         => ( ! [A4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5726676334696518183BT_rat(F,A2)),groups5726676334696518183BT_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8608_prod__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B3)
     => ( aa_set_int_o(ord_less_eq_set_int(A2),B3)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B2)) )
         => ( ! [A4: int] :
                ( aa_set_int_o(member_int(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8609_prod__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A2),B3)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B2)) )
         => ( ! [A4: complex] :
                ( aa_set_complex_o(member_complex(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8610_prod__mono2,axiom,
    ! [B3: set_Code_integer,A2: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A2),B3)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,B2)) )
         => ( ! [A4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A4),A2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,A4)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2555765274223993564er_rat(F,A2)),groups2555765274223993564er_rat(F,B3)) ) ) ) ) ).

% prod_mono2
tff(fact_8611_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ( aa_VEBT_VEBT_complex(F,A) != zero_zero_complex )
       => ( groups127312072573709053omplex(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups127312072573709053omplex(F,A2)),aa_VEBT_VEBT_complex(F,A)),groups127312072573709053omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8612_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( aa_complex_complex(F,A) != zero_zero_complex )
       => ( groups3708469109370488835omplex(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups3708469109370488835omplex(F,A2)),aa_complex_complex(F,A)),groups3708469109370488835omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8613_prod__diff1,axiom,
    ! [A2: set_Code_integer,F: fun_Co1900716531650368264omplex,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( aa_Cod5621350732091595593omplex(F,A) != zero_zero_complex )
       => ( groups862514429393162674omplex(F,minus_2355218937544613996nteger(A2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups862514429393162674omplex(F,A2)),aa_Cod5621350732091595593omplex(F,A)),groups862514429393162674omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8614_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_complex,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( aa_real_complex(F,A) != zero_zero_complex )
       => ( groups713298508707869441omplex(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups713298508707869441omplex(F,A2)),aa_real_complex(F,A)),groups713298508707869441omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8615_prod__diff1,axiom,
    ! [A2: set_int,F: fun_int_complex,A: int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( aa_int_complex(F,A) != zero_zero_complex )
       => ( groups7440179247065528705omplex(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups7440179247065528705omplex(F,A2)),aa_int_complex(F,A)),groups7440179247065528705omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8616_prod__diff1,axiom,
    ! [A2: set_nat,F: fun_nat_complex,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_nat_complex(F,A) != zero_zero_complex )
       => ( groups6464643781859351333omplex(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A2),aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(F,A2)),aa_nat_complex(F,A)),groups6464643781859351333omplex(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8617_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ( aa_VEBT_VEBT_real(F,A) != zero_zero_real )
       => ( groups2703838992350267259T_real(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups2703838992350267259T_real(F,A2)),aa_VEBT_VEBT_real(F,A)),groups2703838992350267259T_real(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8618_prod__diff1,axiom,
    ! [A2: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( aa_complex_real(F,A) != zero_zero_real )
       => ( groups766887009212190081x_real(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups766887009212190081x_real(F,A2)),aa_complex_real(F,A)),groups766887009212190081x_real(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8619_prod__diff1,axiom,
    ! [A2: set_Code_integer,F: fun_Co3960896333048504454r_real,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( aa_Code_integer_real(F,A) != zero_zero_real )
       => ( groups9004974159866482096r_real(F,minus_2355218937544613996nteger(A2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups9004974159866482096r_real(F,A2)),aa_Code_integer_real(F,A)),groups9004974159866482096r_real(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8620_prod__diff1,axiom,
    ! [A2: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( aa_real_real(F,A) != zero_zero_real )
       => ( groups1681761925125756287l_real(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups1681761925125756287l_real(F,A2)),aa_real_real(F,A)),groups1681761925125756287l_real(F,A2)) ) ) ) ).

% prod_diff1
tff(fact_8621_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_8622_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: fun_real_real,Bound: real,I: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_8623_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_o,F: fun_o_real,Bound: real,I: real] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(X))),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_o_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_o(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_8624_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Bound: real,I: real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_8625_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,F: fun_int_real,Bound: real,I: real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_int(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_8626_ln__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),numeral_numeral_real(bit0(one)))
       => ( aa_real_real(ln_ln_real,Xd) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_iv,Xd)) ) ) ) ).

% ln_series
tff(fact_8627_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,C: real,G: fun_VEBT_VEBT_real,D: real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_VEBT_VEBT_real(G,X))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_8628_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,C: real,G: fun_nat_real,D: real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_nat_real(G,X))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_8629_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,C: real,G: fun_real_real,D: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(G,X))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_8630_f__g__map__foldr__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,C: real,G: fun_int_real,D: real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_int_real(G,X))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_8631_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_8632_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_8633_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_o,F: fun_o_nat,Bound: nat,I: nat] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X))),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_o_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_8634_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_8635_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,F: fun_int_nat,Bound: nat,I: nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_int_nat(F,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_8636_map__ident,axiom,
    ! [X4: list_nat] : map_nat_nat(aTP_Lamm_ap,X4) = X4 ).

% map_ident
tff(fact_8637_listsum__bound,axiom,
    ! [Xs: list_complex,F: fun_complex_real,Y: real] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),set_complex2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_complex_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_8638_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Y: real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_8639_listsum__bound,axiom,
    ! [Xs: list_nat,F: fun_nat_real,Y: real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_8640_listsum__bound,axiom,
    ! [Xs: list_real,F: fun_real_real,Y: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_8641_listsum__bound,axiom,
    ! [Xs: list_int,F: fun_int_real,Y: real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F,Xs)),Y)) ) ).

% listsum_bound
tff(fact_8642_length__map,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_8643_length__map,axiom,
    ! [F: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_8644_length__map,axiom,
    ! [F: fun_o_real,Xs: list_o] : size_size_list_real(map_o_real(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_8645_length__map,axiom,
    ! [F: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_8646_length__map,axiom,
    ! [F: fun_int_real,Xs: list_int] : size_size_list_real(map_int_real(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_8647_length__map,axiom,
    ! [F: fun_real_o,Xs: list_real] : size_size_list_o(map_real_o(F,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_8648_length__map,axiom,
    ! [F: fun_o_o,Xs: list_o] : size_size_list_o(map_o_o(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_8649_length__map,axiom,
    ! [F: fun_nat_o,Xs: list_nat] : size_size_list_o(map_nat_o(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_8650_length__map,axiom,
    ! [F: fun_int_o,Xs: list_int] : size_size_list_o(map_int_o(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_8651_length__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : size_size_list_nat(map_VEBT_VEBT_nat(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_8652_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) )
    <=> ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X5) = aa_VEBT_VEBT_nat(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_8653_map__eq__conv,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) )
    <=> ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X5) = aa_VEBT_VEBT_real(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_8654_map__eq__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_nat_nat] :
      ( ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => ( aa_nat_nat(F,X5) = aa_nat_nat(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_8655_map__eq__conv,axiom,
    ! [F: fun_nat_o,Xs: list_nat,G: fun_nat_o] :
      ( ( map_nat_o(F,Xs) = map_nat_o(G,Xs) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Xs))
         => ( aa_nat_o(F,X5)
          <=> aa_nat_o(G,X5) ) ) ) ).

% map_eq_conv
tff(fact_8656_map__replicate,axiom,
    ! [F: fun_nat_nat,Nb: nat,Xd: nat] : map_nat_nat(F,replicate_nat(Nb,Xd)) = replicate_nat(Nb,aa_nat_nat(F,Xd)) ).

% map_replicate
tff(fact_8657_map__replicate,axiom,
    ! [F: fun_nat_o,Nb: nat,Xd: nat] : map_nat_o(F,replicate_nat(Nb,Xd)) = replicate_o(Nb,aa_nat_o(F,Xd)) ).

% map_replicate
tff(fact_8658_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_nat,Nb: nat,Xd: vEBT_VEBT] : map_VEBT_VEBT_nat(F,replicate_VEBT_VEBT(Nb,Xd)) = replicate_nat(Nb,aa_VEBT_VEBT_nat(F,Xd)) ).

% map_replicate
tff(fact_8659_map__replicate,axiom,
    ! [F: fun_VEBT_VEBT_real,Nb: nat,Xd: vEBT_VEBT] : map_VEBT_VEBT_real(F,replicate_VEBT_VEBT(Nb,Xd)) = replicate_real(Nb,aa_VEBT_VEBT_real(F,Xd)) ).

% map_replicate
tff(fact_8660_map__replicate,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Nb: nat,Xd: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,replicate_VEBT_VEBT(Nb,Xd)) = replicate_VEBT_VEBT(Nb,aa_VEB5832953974749077948T_VEBT(F,Xd)) ).

% map_replicate
tff(fact_8661_real__nat__list,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_iw(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_8662_real__nat__list,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(aTP_Lamm_ix(F),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_8663_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),Nb) = aa_VEB5832953974749077948T_VEBT(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8664_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VE9120655849212850072T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),Nb) = aa_VEB4941942288398148609T_VEBT(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8665_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBTi_nat(F,Xs)),Nb) = aa_VEBT_VEBTi_nat(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8666_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VE712148701811721852_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),Nb) = aa_VEB3973023575446460973_VEBTi(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8667_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VE8449866691444886503_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),Nb) = aa_VEB5586453056816421800_VEBTi(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8668_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(F,Xs)),Nb) = aa_VEBT_VEBT_int(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8669_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,F: fun_VEBT_VEBTi_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBTi_int(F,Xs)),Nb) = aa_VEBT_VEBTi_int(F,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8670_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),Nb) = aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8671_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F,Xs)),Nb) = aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% nth_map
tff(fact_8672_nth__map,axiom,
    ! [Nb: nat,Xs: list_real,F: fun_real_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( nth_VEBT_VEBT(map_real_VEBT_VEBT(F,Xs),Nb) = aa_real_VEBT_VEBT(F,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% nth_map
tff(fact_8673_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_iy(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_8674_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_iz(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_8675_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8676_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8677_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8678_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8679_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_o_nat,Ys: list_o] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_o_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8680_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_o_real,Ys: list_o] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_o_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8681_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8682_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8683_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_int_nat,Ys: list_int] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_int_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8684_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_int_real,Ys: list_int] :
      ( ( map_VEBT_VEBT_real(F,Xs) = map_int_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_8685_map__update,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,K: nat,Y: nat] : map_nat_nat(F,list_update_nat(Xs,K,Y)) = list_update_nat(map_nat_nat(F,Xs),K,aa_nat_nat(F,Y)) ).

% map_update
tff(fact_8686_map__update,axiom,
    ! [F: fun_nat_o,Xs: list_nat,K: nat,Y: nat] : map_nat_o(F,list_update_nat(Xs,K,Y)) = list_update_o(map_nat_o(F,Xs),K,aa_nat_o(F,Y)) ).

% map_update
tff(fact_8687_map__update,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_update_nat(map_VEBT_VEBT_nat(F,Xs),K,aa_VEBT_VEBT_nat(F,Y)) ).

% map_update
tff(fact_8688_map__update,axiom,
    ! [F: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_update_real(map_VEBT_VEBT_real(F,Xs),K,aa_VEBT_VEBT_real(F,Y)) ).

% map_update
tff(fact_8689_map__update,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_u1324408373059187874T_VEBT(map_VE8901447254227204932T_VEBT(F,Xs),K,aa_VEB5832953974749077948T_VEBT(F,Y)) ).

% map_update
tff(fact_8690_map__update,axiom,
    ! [F: fun_VE712148701811721852_VEBTi,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] : map_VE7029150624388687525_VEBTi(F,list_u1324408373059187874T_VEBT(Xs,K,Y)) = list_u6098035379799741383_VEBTi(map_VE7029150624388687525_VEBTi(F,Xs),K,aa_VEB3973023575446460973_VEBTi(F,Y)) ).

% map_update
tff(fact_8691_map__update,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] : map_VE7998069337340375161T_VEBT(F,list_u6098035379799741383_VEBTi(Xs,K,Y)) = list_u1324408373059187874T_VEBT(map_VE7998069337340375161T_VEBT(F,Xs),K,aa_VEB4941942288398148609T_VEBT(F,Y)) ).

% map_update
tff(fact_8692_map__update,axiom,
    ! [F: fun_VE8449866691444886503_VEBTi,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] : map_VE483055756984248624_VEBTi(F,list_u6098035379799741383_VEBTi(Xs,K,Y)) = list_u6098035379799741383_VEBTi(map_VE483055756984248624_VEBTi(F,Xs),K,aa_VEB5586453056816421800_VEBTi(F,Y)) ).

% map_update
tff(fact_8693_list_Omap__ident,axiom,
    ! [T: list_nat] : map_nat_nat(aTP_Lamm_ap,T) = T ).

% list.map_ident
tff(fact_8694_list_Omap__cong,axiom,
    ! [Xd: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xd = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
       => ( map_VEBT_VEBT_nat(F,Xd) = map_VEBT_VEBT_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_8695_list_Omap__cong,axiom,
    ! [Xd: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xd = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
       => ( map_VEBT_VEBT_real(F,Xd) = map_VEBT_VEBT_real(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_8696_list_Omap__cong,axiom,
    ! [Xd: list_nat,Ya: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xd = Ya )
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(member_nat(Z2),set_nat2(Ya))
           => ( aa_nat_nat(F,Z2) = aa_nat_nat(G,Z2) ) )
       => ( map_nat_nat(F,Xd) = map_nat_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_8697_list_Omap__cong,axiom,
    ! [Xd: list_nat,Ya: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ( Xd = Ya )
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(member_nat(Z2),set_nat2(Ya))
           => ( aa_nat_o(F,Z2)
            <=> aa_nat_o(G,Z2) ) )
       => ( map_nat_o(F,Xd) = map_nat_o(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_8698_list_Omap__cong0,axiom,
    ! [Xd: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xd))
         => ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
     => ( map_VEBT_VEBT_nat(F,Xd) = map_VEBT_VEBT_nat(G,Xd) ) ) ).

% list.map_cong0
tff(fact_8699_list_Omap__cong0,axiom,
    ! [Xd: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xd))
         => ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
     => ( map_VEBT_VEBT_real(F,Xd) = map_VEBT_VEBT_real(G,Xd) ) ) ).

% list.map_cong0
tff(fact_8700_list_Omap__cong0,axiom,
    ! [Xd: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),set_nat2(Xd))
         => ( aa_nat_nat(F,Z2) = aa_nat_nat(G,Z2) ) )
     => ( map_nat_nat(F,Xd) = map_nat_nat(G,Xd) ) ) ).

% list.map_cong0
tff(fact_8701_list_Omap__cong0,axiom,
    ! [Xd: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),set_nat2(Xd))
         => ( aa_nat_o(F,Z2)
          <=> aa_nat_o(G,Z2) ) )
     => ( map_nat_o(F,Xd) = map_nat_o(G,Xd) ) ) ).

% list.map_cong0
tff(fact_8702_list_Oinj__map__strong,axiom,
    ! [Xd: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Fa: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xd))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_nat(F,Z2) = aa_VEBT_VEBT_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_nat(F,Xd) = map_VEBT_VEBT_nat(Fa,Xaa) )
       => ( Xd = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_8703_list_Oinj__map__strong,axiom,
    ! [Xd: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Fa: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xd))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_real(F,Z2) = aa_VEBT_VEBT_real(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_real(F,Xd) = map_VEBT_VEBT_real(Fa,Xaa) )
       => ( Xd = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_8704_list_Oinj__map__strong,axiom,
    ! [Xd: list_nat,Xaa: list_nat,F: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [Z2: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z2),set_nat2(Xd))
         => ( aa_set_nat_o(member_nat(Za),set_nat2(Xaa))
           => ( ( aa_nat_nat(F,Z2) = aa_nat_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_nat_nat(F,Xd) = map_nat_nat(Fa,Xaa) )
       => ( Xd = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_8705_list_Oinj__map__strong,axiom,
    ! [Xd: list_nat,Xaa: list_nat,F: fun_nat_o,Fa: fun_nat_o] :
      ( ! [Z2: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z2),set_nat2(Xd))
         => ( aa_set_nat_o(member_nat(Za),set_nat2(Xaa))
           => ( ( aa_nat_o(F,Z2)
              <=> aa_nat_o(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_nat_o(F,Xd) = map_nat_o(Fa,Xaa) )
       => ( Xd = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_8706_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F,X) = aa_VEBT_VEBT_nat(G,X) ) )
     => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Xs) ) ) ).

% map_ext
tff(fact_8707_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F,X) = aa_VEBT_VEBT_real(G,X) ) )
     => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Xs) ) ) ).

% map_ext
tff(fact_8708_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_nat_nat(F,X) = aa_nat_nat(G,X) ) )
     => ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) ) ) ).

% map_ext
tff(fact_8709_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_nat_o(F,X)
          <=> aa_nat_o(G,X) ) )
     => ( map_nat_o(F,Xs) = map_nat_o(G,Xs) ) ) ).

% map_ext
tff(fact_8710_map__idI,axiom,
    ! [Xs: list_complex,F: fun_complex_complex] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),set_complex2(Xs))
         => ( aa_complex_complex(F,X) = X ) )
     => ( map_complex_complex(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_8711_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X) = X ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_8712_map__idI,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => ( aa_nat_nat(F,X) = X ) )
     => ( map_nat_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_8713_map__idI,axiom,
    ! [Xs: list_real,F: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => ( aa_real_real(F,X) = X ) )
     => ( map_real_real(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_8714_map__idI,axiom,
    ! [Xs: list_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => ( aa_int_int(F,X) = X ) )
     => ( map_int_int(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_8715_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xs = Ys )
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_nat(F,X) = aa_VEBT_VEBT_nat(G,X) ) )
       => ( map_VEBT_VEBT_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_8716_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xs = Ys )
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_real(F,X) = aa_VEBT_VEBT_real(G,X) ) )
       => ( map_VEBT_VEBT_real(F,Xs) = map_VEBT_VEBT_real(G,Ys) ) ) ) ).

% map_cong
tff(fact_8717_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xs = Ys )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Ys))
           => ( aa_nat_nat(F,X) = aa_nat_nat(G,X) ) )
       => ( map_nat_nat(F,Xs) = map_nat_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_8718_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_o,G: fun_nat_o] :
      ( ( Xs = Ys )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Ys))
           => ( aa_nat_o(F,X)
            <=> aa_nat_o(G,X) ) )
       => ( map_nat_o(F,Xs) = map_nat_o(G,Ys) ) ) ) ).

% map_cong
tff(fact_8719_ex__map__conv,axiom,
    ! [Ys: list_o,F: fun_nat_o] :
      ( ? [Xs4: list_nat] : Ys = map_nat_o(F,Xs4)
    <=> ! [X5: $o] :
          ( member_o((X5),set_o2(Ys))
         => ? [Xa3: nat] :
              ( (X5)
            <=> aa_nat_o(F,Xa3) ) ) ) ).

% ex_map_conv
tff(fact_8720_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_VEBT_VEBT_nat] :
      ( ? [Xs4: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_nat(F,Xs4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Ys))
         => ? [Xa3: vEBT_VEBT] : X5 = aa_VEBT_VEBT_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_8721_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_nat_nat] :
      ( ? [Xs4: list_nat] : Ys = map_nat_nat(F,Xs4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_nat2(Ys))
         => ? [Xa3: nat] : X5 = aa_nat_nat(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_8722_ex__map__conv,axiom,
    ! [Ys: list_real,F: fun_VEBT_VEBT_real] :
      ( ? [Xs4: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_real(F,Xs4)
    <=> ! [X5: real] :
          ( aa_set_real_o(member_real(X5),set_real2(Ys))
         => ? [Xa3: vEBT_VEBT] : X5 = aa_VEBT_VEBT_real(F,Xa3) ) ) ).

% ex_map_conv
tff(fact_8723_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_ja(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_8724_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_jb(K),Lst) = replicate_real(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_8725_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_jc(K),Lst) = replicate_VEBT_VEBT(size_size_list_real(Lst),K) ).

% map_replicate_const
tff(fact_8726_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] : map_o_VEBT_VEBT(aTP_Lamm_jd(K),Lst) = replicate_VEBT_VEBT(size_size_list_o(Lst),K) ).

% map_replicate_const
tff(fact_8727_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_je,K),Lst) = replicate_nat(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_8728_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] : map_nat_o(aa_o_fun_nat_o(aTP_Lamm_jf,(K)),Lst) = replicate_o(size_size_list_nat(Lst),(K)) ).

% map_replicate_const
tff(fact_8729_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_jg(K),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_8730_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_jh(K),Lst) = replicate_VEBT_VEBT(size_size_list_int(Lst),K) ).

% map_replicate_const
tff(fact_8731_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat,Xd: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_nat(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(F,Xd) ) )
     => ( map_VEBT_VEBT_nat(F,list_u1324408373059187874T_VEBT(L,I,Xd)) = map_VEBT_VEBT_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_8732_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: fun_VEBT_VEBT_real,Xd: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_real(F,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(F,Xd) ) )
     => ( map_VEBT_VEBT_real(F,list_u1324408373059187874T_VEBT(L,I,Xd)) = map_VEBT_VEBT_real(F,L) ) ) ).

% map_upd_eq
tff(fact_8733_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_nat,Xd: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(F,Xd) ) )
     => ( map_nat_nat(F,list_update_nat(L,I,Xd)) = map_nat_nat(F,L) ) ) ).

% map_upd_eq
tff(fact_8734_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: fun_nat_o,Xd: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_o(F,aa_nat_nat(nth_nat(L),I))
        <=> aa_nat_o(F,Xd) ) )
     => ( map_nat_o(F,list_update_nat(L,I,Xd)) = map_nat_o(F,L) ) ) ).

% map_upd_eq
tff(fact_8735_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_8736_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_8737_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xd) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_8738_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xd) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_8739_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_8740_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xd) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != numeral_numeral_nat(bit0(bit0(one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_8741_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = 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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_8742_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xd) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != numeral_numeral_nat(aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != 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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_8743_arctan__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => ( aa_real_real(arctan,Xd) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ji,Xd)) ) ) ).

% arctan_series
tff(fact_8744_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_8745_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_8746_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_jj) = zero_zero_complex ).

% suminf_zero
tff(fact_8747_suminf__zero,axiom,
    suminf_real(aTP_Lamm_jk) = zero_zero_real ).

% suminf_zero
tff(fact_8748_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_av) = zero_zero_nat ).

% suminf_zero
tff(fact_8749_suminf__zero,axiom,
    suminf_int(aTP_Lamm_jl) = zero_zero_int ).

% suminf_zero
tff(fact_8750_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) = suminf_real(aTP_Lamm_jm) ).

% pi_series
tff(fact_8751_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,Xd: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),Xd)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),list_u6098035379799741383_VEBTi(Tree_is,H,Xd)))) ) ).

% big_assn_simp
tff(fact_8752_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or66887138388493659n_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_real,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8753_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_set_nat(I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8754_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or4029947393144176647an_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_rat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8755_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or1222409239386451017an_num(L,U))
    <=> ( aa_num_o(ord_less_eq_num(L),I)
        & aa_num_o(ord_less_num(I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8756_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or4665077453230672383an_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_nat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8757_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),set_or8404916559141939852nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8758_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or4662586982721622107an_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_int,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_8759_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8760_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8761_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ aa_num_o(ord_less_num(A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8762_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8763_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8764_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_8765_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or66887138388493659n_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8766_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8767_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(ord_less_num(A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8768_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8769_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8770_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_8771_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Ico_iff
tff(fact_8772_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Ico_iff
tff(fact_8773_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y)))) ) ).

% local.ext
tff(fact_8774_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% recomp
tff(fact_8775_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% repack
tff(fact_8776_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(Y,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% txe
tff(fact_8777_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_8778_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_real,Y: vEBT_VEBT,Xd: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_real(TreeList4) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xd)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))) ) ) ).

% tcd
tff(fact_8779_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_o,Y: vEBT_VEBT,Xd: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_o(TreeList4) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xd)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))) ) ) ).

% tcd
tff(fact_8780_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_nat,Y: vEBT_VEBT,Xd: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_nat(TreeList4) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xd)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))) ) ) ).

% tcd
tff(fact_8781_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_int,Y: vEBT_VEBT,Xd: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_int(TreeList4) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xd)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xd)))) ) ) ).

% tcd
tff(fact_8782_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xd: vEBT_VEBTi,Xba: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( Xaa = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xd)),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xd))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,Xaa),list_u6098035379799741383_VEBTi(Tree_is,H,Xd)))) ) ) ).

% big_assn_simp'
tff(fact_8783_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_rat(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8784_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_assn,M: nat,Nb: nat] :
      groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_assn,aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_assn(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8785_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Nb: nat] :
      groups129246275422532515t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_real(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8786_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_Code_integer,M: nat,Nb: nat] :
      groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_Code_integer(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8787_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_complex,M: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_complex(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8788_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_nat(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8789_prod_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M),one_one_int,aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_nat_int(G,Nb))) ).

% prod.op_ivl_Suc
tff(fact_8790_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8791_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8792_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( aa_num_o(ord_less_num(A),B)
     => ( aa_num_o(ord_less_num(C),D)
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8793_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8794_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8795_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
        <=> ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_8796_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8797_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8798_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( aa_num_o(ord_less_num(A),B)
       => ( aa_num_o(ord_less_num(C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8799_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8800_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8801_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_8802_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C,D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8803_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8804_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C,D) )
     => ( aa_num_o(ord_less_num(A),B)
       => ( aa_num_o(ord_less_num(C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8805_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C),D) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8806_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C,D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8807_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C,D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_8808_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_8809_listI__assn__conv,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( Nb = size_s6755466524823107622T_VEBT(Xs) )
     => ( vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb),A2,Xs,Xsi) = vEBT_L6296928887356842470_VEBTi(A2,Xs,Xsi) ) ) ).

% listI_assn_conv
tff(fact_8810_list__assn__conv__idx,axiom,
    ! [A2: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : vEBT_L6296928887356842470_VEBTi(A2,Xs,Xsi) = vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),A2,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_8811_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A,B)) ) ).

% infinite_Ico
tff(fact_8812_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B)) ) ).

% infinite_Ico
tff(fact_8813_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
         => aa_nat_o(P,M4) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X5) ) ) ).

% all_nat_less_eq
tff(fact_8814_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
          & aa_nat_o(P,M4) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X5) ) ) ).

% ex_nat_less_eq
tff(fact_8815_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_8816_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_8817_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_8818_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8819_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_8820_listI__assn__conv_H,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,A2: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F2: assn] :
      ( ( Nb = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(times_times_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb),A2,Xs,Xsi)),F2) = aa_assn_assn(times_times_assn(vEBT_L6296928887356842470_VEBTi(A2,Xs,Xsi)),F2) ) ) ).

% listI_assn_conv'
tff(fact_8821_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_di(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_8822_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_dj(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_8823_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk(G),K)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) ).

% prod.shift_bounds_nat_ivl
tff(fact_8824_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_dl(G),K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) ).

% prod.shift_bounds_nat_ivl
tff(fact_8825_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D)
               => ( aa_nat_nat(G,X) = aa_nat_nat(H,X) ) ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = aa_set_nat_nat(groups708209901874060359at_nat(H),aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8826_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: fun_nat_int,H: fun_nat_int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D)
               => ( aa_nat_int(G,X) = aa_nat_int(H,X) ) ) )
         => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = groups705719431365010083at_int(H,aa_nat_set_nat(set_or4665077453230672383an_nat(C),D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8827_prod_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D: int,G: fun_int_int,H: fun_int_int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),X)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),D)
               => ( aa_int_int(G,X) = aa_int_int(H,X) ) ) )
         => ( groups1705073143266064639nt_int(G,set_or4662586982721622107an_int(A,B)) = groups1705073143266064639nt_int(H,set_or4662586982721622107an_int(C,D)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_8828_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_assn_assn(times_times_assn(groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = groups6906906614972039071t_assn(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8829_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = groups129246275422532515t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8830_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = groups3455450783089532116nteger(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8831_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),groups6464643781859351333omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = groups6464643781859351333omplex(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8832_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8833_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,Nb: nat,P3: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),P3)
       => ( aa_int_int(times_times_int(groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),P3))) = groups705719431365010083at_int(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_8834_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_8835_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8836_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% pi_less_4
tff(fact_8837_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),pi) ).

% pi_ge_two
tff(fact_8838_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) != numeral_numeral_real(bit0(one)) ).

% pi_half_neq_two
tff(fact_8839_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_8840_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_8841_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_8842_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_8843_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% pi_half_less_two
tff(fact_8844_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% pi_half_le_two
tff(fact_8845_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),numeral_numeral_nat(K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),pred_numeral(K)),insert_nat(pred_numeral(K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),pred_numeral(K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_8846_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ).

% pi_half_gt_zero
tff(fact_8847_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ).

% pi_half_ge_zero
tff(fact_8848_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8849_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ).

% arctan_ubound
tff(fact_8850_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% arctan_one
tff(fact_8851_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8852_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ).

% arctan_bounded
tff(fact_8853_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_8854_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,bit0(one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(bit0(one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% machin_Euler
tff(fact_8855_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(bit0(one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,bit0(one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,bit0(aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_8856_arctan__inverse,axiom,
    ! [Xd: real] :
      ( ( Xd != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xd)) = 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,Xd)),pi)),numeral_numeral_real(bit0(one)))),aa_real_real(arctan,Xd)) ) ) ).

% arctan_inverse
tff(fact_8857_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,numeral_numeral_nat(bit0(one))),Nb)))),pi)),numeral_numeral_real(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_8858_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_8859_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_8860_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_8861_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_8862_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_8863_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = one_one_real ).

% sin_pi_half
tff(fact_8864_sin__periodic,axiom,
    ! [Xd: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(sin_real,Xd) ).

% sin_periodic
tff(fact_8865_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8866_sin__2pi__minus,axiom,
    ! [Xd: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xd)) ).

% sin_2pi_minus
tff(fact_8867_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8868_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(bit0(one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_8869_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_8870_sin__x__le__x,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xd)),Xd) ) ).

% sin_x_le_x
tff(fact_8871_sin__le__one,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xd)),one_one_real) ).

% sin_le_one
tff(fact_8872_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_8873_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,plus_p5714425477246183910nteger(U,one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_8874_sin__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xd)) ) ) ).

% sin_gt_zero
tff(fact_8875_sin__x__ge__neg__x,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xd)),aa_real_real(sin_real,Xd)) ) ).

% sin_x_ge_neg_x
tff(fact_8876_sin__ge__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xd)) ) ) ).

% sin_ge_zero
tff(fact_8877_sin__ge__minus__one,axiom,
    ! [Xd: 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,Xd)) ).

% sin_ge_minus_one
tff(fact_8878_abs__sin__le__one,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xd))),one_one_real) ).

% abs_sin_le_one
tff(fact_8879_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(plus_plus_int(U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_8880_sin__eq__0__pi,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),pi)
       => ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
         => ( Xd = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8881_divmod__int__def,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(M,Nb) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_8882_sin__zero__pi__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),pi)
     => ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
      <=> ( Xd = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8883_sin__zero__iff__int2,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
    <=> ? [I2: int] : Xd = 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_8884_sin__gt__zero__02,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),numeral_numeral_real(bit0(one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xd)) ) ) ).

% sin_gt_zero_02
tff(fact_8885_divmod_H__nat__def,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(M,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb))) ).

% divmod'_nat_def
tff(fact_8886_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_8887_sin__gt__zero2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xd)) ) ) ).

% sin_gt_zero2
tff(fact_8888_sin__lt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xd)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8889_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))) ).

% sin_30
tff(fact_8890_sin__inj__pi,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( ( aa_real_real(sin_real,Xd) = aa_real_real(sin_real,Y) )
             => ( Xd = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8891_sin__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xd)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8892_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(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,Xd)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8893_sin__le__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xd)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8894_sin__less__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xd)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8895_sin__monotone__2pi,axiom,
    ! [Y: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xd)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8896_sin__mono__less__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xd)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8897_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)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
            & ( aa_real_real(sin_real,X) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
                  & ( aa_real_real(sin_real,Y3) = Y ) )
               => ( Y3 = X ) ) ) ) ) ).

% sin_total
tff(fact_8898_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(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_8899_sin__zero__iff__int,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
    <=> ? [I2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I2)
          & ( Xd = 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),numeral_numeral_real(bit0(one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8900_sin__zero__lemma,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
            & ( Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8901_sin__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N6)
            & ( Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N6)
            & ( Xd = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8902_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_8903_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_8904_summable__arctan__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ji,Xd)) ) ).

% summable_arctan_series
tff(fact_8905_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(product_Pair_int_int(Q2,R2)) = aa_int_int(plus_plus_int(Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_8906_minus__numeral__div__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Nb))) ).

% minus_numeral_div_numeral
tff(fact_8907_numeral__div__minus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Nb))) ).

% numeral_div_minus_numeral
tff(fact_8908_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: 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),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_jn(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8909_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( eucl_rel_int(aa_int_int(plus_plus_int(A),one_one_int),B,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B),product_Pair_int_int(Q2,aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_8910_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),A)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),B),product_Pair_int_int(Q2,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_8911_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M))))),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8912_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_8913_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_8914_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_8915_cos__periodic,axiom,
    ! [Xd: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cos_real,Xd) ).

% cos_periodic
tff(fact_8916_cos__2pi__minus,axiom,
    ! [Xd: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),Xd)) = aa_real_real(cos_real,Xd) ).

% cos_2pi_minus
tff(fact_8917_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_8918_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_8919_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8920_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8921_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(bit0(one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_8922_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,bit0(one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8923_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q6,R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_8924_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q6,R4))
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
tff(fact_8925_cos__le__one,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xd)),one_one_real) ).

% cos_le_one
tff(fact_8926_polar__Ex,axiom,
    ! [Xd: real,Y: real] :
    ? [R3: real,A4: real] :
      ( ( Xd = 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_8927_cos__arctan__not__zero,axiom,
    ! [Xd: real] : aa_real_real(cos_real,aa_real_real(arctan,Xd)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_8928_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xd)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8929_cos__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xd) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8930_cos__inj__pi,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd) = aa_real_real(cos_real,Y) )
             => ( Xd = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8931_cos__ge__minus__one,axiom,
    ! [Xd: 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,Xd)) ).

% cos_ge_minus_one
tff(fact_8932_abs__cos__le__one,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xd))),one_one_real) ).

% abs_cos_le_one
tff(fact_8933_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_8934_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_8935_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(divide_divide_int(K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_8936_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(bit0(one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_8937_cos__monotone__0__pi,axiom,
    ! [Y: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xd)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8938_cos__mono__less__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8939_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8940_sin__zero__abs__cos__one,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sin_real,Xd) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xd)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_8941_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(times_times_int(Q2),L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q2,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_8942_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(divide_divide_int(K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_8943_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8944_cos__is__zero,axiom,
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),numeral_numeral_real(bit0(one)))
      & ( aa_real_real(cos_real,X) = zero_zero_real )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),numeral_numeral_real(bit0(one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X ) ) ) ).

% cos_is_zero
tff(fact_8945_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8946_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xd: 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),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8947_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)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
            & ( aa_real_real(cos_real,X) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
                  & ( aa_real_real(cos_real,Y3) = Y ) )
               => ( Y3 = X ) ) ) ) ) ).

% cos_total
tff(fact_8948_sincos__principal__value,axiom,
    ! [Xd: real] :
    ? [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,Xd) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,Xd) ) ) ).

% sincos_principal_value
tff(fact_8949_sin__cos__le1,axiom,
    ! [Xd: 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,Xd)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xd)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_8950_cos__double__less__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),numeral_numeral_real(bit0(one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),Xd))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8951_cos__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xd)) ) ) ).

% cos_gt_zero
tff(fact_8952_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one))) ).

% cos_60
tff(fact_8953_cos__one__2pi__int,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(cos_real,Xd) = one_one_real )
    <=> ? [X5: int] : Xd = 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,X5)),numeral_numeral_real(bit0(one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8954_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q2,R2))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,
            product_Pair_int_int($ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,Q2)),one_one_int)),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_8955_cos__gt__zero__pi,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xd)) ) ) ).

% cos_gt_zero_pi
tff(fact_8956_cos__ge__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xd)) ) ) ).

% cos_ge_zero
tff(fact_8957_cos__one__2pi,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(cos_real,Xd) = one_one_real )
    <=> ( ? [X5: nat] : Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X5)),numeral_numeral_real(bit0(one)))),pi)
        | ? [X5: nat] : Xd = 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,X5)),numeral_numeral_real(bit0(one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8958_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q2,R2))
    <=> ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(L),Q2)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_8959_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q2),L)),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q2,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8960_sincos__total__pi,axiom,
    ! [Y: real,Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),pi)
            & ( Xd = aa_real_real(cos_real,T5) )
            & ( Y = aa_real_real(sin_real,T5) ) ) ) ) ).

% sincos_total_pi
tff(fact_8961_sin__expansion__lemma,axiom,
    ! [Xd: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one))))) ).

% sin_expansion_lemma
tff(fact_8962_cos__zero__iff__int,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(cos_real,Xd) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I2)
          & ( Xd = 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),numeral_numeral_real(bit0(one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8963_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,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(times_times_int(Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q5,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(plus_plus_int(aa_int_int(times_times_int(Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8964_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) ) )
       => ( ! [Q4: int] :
              ( ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(times_times_int(Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = product_Pair_int_int(Q4,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(plus_plus_int(aa_int_int(times_times_int(Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8965_cos__zero__lemma,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( ( aa_real_real(cos_real,Xd) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
            & ( Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8966_cos__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(cos_real,Xd) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N6)
            & ( Xd = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N6)
            & ( Xd = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8967_cos__expansion__lemma,axiom,
    ! [Xd: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one)))))) ).

% cos_expansion_lemma
tff(fact_8968_sincos__total__pi__half,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
              & ( Xd = aa_real_real(cos_real,T5) )
              & ( Y = aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8969_sincos__total__2pi__le,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))
          & ( Xd = aa_real_real(cos_real,T5) )
          & ( Y = aa_real_real(sin_real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8970_sincos__total__2pi,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
     => ~ ! [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))
             => ( ( Xd = aa_real_real(cos_real,T5) )
               => ( Y != aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8971__C11_C,axiom,
    ! [Xaa: array_VEBT_VEBTi,Xd: list_VEBT_VEBTi] : hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Xaa,Xd)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one)))))),Xd)),vEBT_V739175172307565963ildupi(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one))))),aa_lis5319508155010942429i_assn(aTP_Lamm_jo(Xaa),Xd)) ).

% "11"
tff(fact_8972_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn_Code_integer(K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_8973_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M4: int] :
        ? [N6: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M4),aa_int_int(abs_abs_int,N6))
          & aa_set_int_o(member_int(N6),S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8974_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_8975__C3_OIH_C_I4_J,axiom,
    ! [Xd: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,na)))
     => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one))) )
       => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(aa_nat_nat(suc,Xd)),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa_nat_nat(suc,Xd)))) ) ) ).

% "3.IH"(4)
tff(fact_8976__C3_OIH_C_I3_J,axiom,
    ! [Xd: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,na)))
     => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one))) )
       => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Xd),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Xd))) ) ) ).

% "3.IH"(3)
tff(fact_8977__C3_OIH_C_I1_J,axiom,
    ! [Xd: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,na)))
     => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one))) )
       => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Xd),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(Xd))) ) ) ).

% "3.IH"(1)
tff(fact_8978_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_8979_tan__periodic__n,axiom,
    ! [Xd: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Nb)),pi))) = aa_real_real(tan_real,Xd) ).

% tan_periodic_n
tff(fact_8980_tan__periodic__nat,axiom,
    ! [Xd: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xd) ).

% tan_periodic_nat
tff(fact_8981_tan__periodic__int,axiom,
    ! [Xd: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xd) ).

% tan_periodic_int
tff(fact_8982_tan__periodic,axiom,
    ! [Xd: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(tan_real,Xd) ).

% tan_periodic
tff(fact_8983_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_8984_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_8985_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_8986_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_8987_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_8988_uminus__integer__code_I1_J,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_8989_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_8990_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_8991_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = one_one_real ).

% tan_45
tff(fact_8992_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X)) ) ) ).

% lemma_tan_total
tff(fact_8993_tan__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xd)) ) ) ).

% tan_gt_zero
tff(fact_8994_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,X) = Y ) ) ).

% lemma_tan_total1
tff(fact_8995_tan__mono__lt__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xd)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8996_tan__monotone_H,axiom,
    ! [Y: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xd)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8997_tan__monotone,axiom,
    ! [Y: real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xd)) ) ) ) ).

% tan_monotone
tff(fact_8998_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,X) = Y )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
            & ( aa_real_real(tan_real,Y3) = Y ) )
         => ( Y3 = X ) ) ) ).

% tan_total
tff(fact_8999_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_9000_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),Y)) ).

% tan_inverse
tff(fact_9001_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M4: nat] :
        ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N6)
          & aa_set_nat_o(member_nat(N6),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9002_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N7)
              & aa_set_nat_o(member_nat(N7),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_9003_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
          & ( aa_real_real(tan_real,X) = Y ) ) ) ).

% tan_total_pos
tff(fact_9004_tan__pos__pi2__le,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xd)) ) ) ).

% tan_pos_pi2_le
tff(fact_9005_tan__less__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xd)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_9006_tan__mono__le,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xd)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_9007_tan__mono__le__eq,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xd)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9008_tan__bound__pi2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xd))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_9009_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_9010_arctan__tan,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xd)) = Xd ) ) ) ).

% arctan_tan
tff(fact_9011_arctan__unique,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( ( aa_real_real(tan_real,Xd) = Y )
         => ( aa_real_real(arctan,Y) = Xd ) ) ) ) ).

% arctan_unique
tff(fact_9012_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),uminus1351360451143612070nteger(K),K) ).

% abs_integer_code
tff(fact_9013_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_9014_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9015_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9016_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = uminus1351360451143612070nteger(L) ).

% minus_integer_code(2)
tff(fact_9017_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9018_tan__total__pi4,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one)))))
          & ( aa_real_real(tan_real,Z2) = Xd ) ) ) ).

% tan_total_pi4
tff(fact_9019_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(Nb),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% T_vebt_buildupi
tff(fact_9020_highi__h,axiom,
    ! [Xd: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(Xd,Nb),aa_nat_fun_nat_assn(aTP_Lamm_ce(Xd),Nb)) ).

% highi_h
tff(fact_9021_lowi__h,axiom,
    ! [Xd: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(Xd,Nb),aa_nat_fun_nat_assn(aTP_Lamm_cd(Xd),Nb)) ).

% lowi_h
tff(fact_9022_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),
        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(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),code_integer_of_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9023_sin__tan,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
     => ( aa_real_real(sin_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xd)),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,Xd)),numeral_numeral_nat(bit0(one)))))) ) ) ).

% sin_tan
tff(fact_9024_cos__tan,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
     => ( aa_real_real(cos_real,Xd) = 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,Xd)),numeral_numeral_nat(bit0(one)))))) ) ) ).

% cos_tan
tff(fact_9025_real__sqrt__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_real(sqrt,Xd) = aa_real_real(sqrt,Y) )
    <=> ( Xd = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_9026_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sqrt,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_9027_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_9028_real__sqrt__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% real_sqrt_less_iff
tff(fact_9029_real__sqrt__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ).

% real_sqrt_le_iff
tff(fact_9030_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_9031_real__sqrt__eq__1__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sqrt,Xd) = one_one_real )
    <=> ( Xd = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_9032_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_9033_real__sqrt__lt__0__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_9034_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_9035_real__sqrt__le__0__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_9036_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_9037_real__sqrt__lt__1__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xd)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_9038_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_9039_real__sqrt__le__1__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_9040_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_9041_real__sqrt__abs2,axiom,
    ! [Xd: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd)) = aa_real_real(abs_abs_real,Xd) ).

% real_sqrt_abs2
tff(fact_9042_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(bit0(bit0(one)))) = numeral_numeral_real(bit0(one)) ).

% real_sqrt_four
tff(fact_9043_real__sqrt__abs,axiom,
    ! [Xd: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))) = aa_real_real(abs_abs_real,Xd) ).

% real_sqrt_abs
tff(fact_9044_real__sqrt__pow2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xd)),numeral_numeral_nat(bit0(one))) = Xd ) ) ).

% real_sqrt_pow2
tff(fact_9045_real__sqrt__pow2__iff,axiom,
    ! [Xd: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xd)),numeral_numeral_nat(bit0(one))) = Xd )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% real_sqrt_pow2_iff
tff(fact_9046_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))))),numeral_numeral_nat(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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_9047_real__sqrt__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_9048_real__sqrt__mult,axiom,
    ! [Xd: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_9049_real__sqrt__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_9050_real__sqrt__divide,axiom,
    ! [Xd: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_9051_real__sqrt__power,axiom,
    ! [Xd: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xd)),K) ).

% real_sqrt_power
tff(fact_9052_divmod__integer_H__def,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(M,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_9053_real__sqrt__minus,axiom,
    ! [Xd: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xd)) ).

% real_sqrt_minus
tff(fact_9054_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xd: int] : bit_se2793503036327961859nteger(Xaa,code_integer_of_int(Xd)) = code_integer_of_int(aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xaa),Xd)) ).

% Code_Numeral.set_bit_integer.abs_eq
tff(fact_9055_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] : divide6298287555418463151nteger(code_integer_of_int(Xaa),code_integer_of_int(Xd)) = code_integer_of_int(aa_int_int(divide_divide_int(Xaa),Xd)) ).

% divide_integer.abs_eq
tff(fact_9056_modulo__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] : modulo364778990260209775nteger(code_integer_of_int(Xaa),code_integer_of_int(Xd)) = code_integer_of_int(aa_int_int(modulo_modulo_int(Xaa),Xd)) ).

% modulo_integer.abs_eq
tff(fact_9057_real__sqrt__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xd)) ) ).

% real_sqrt_gt_zero
tff(fact_9058_real__sqrt__ge__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xd)) ) ).

% real_sqrt_ge_zero
tff(fact_9059_real__sqrt__eq__zero__cancel,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( ( aa_real_real(sqrt,Xd) = zero_zero_real )
       => ( Xd = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9060_real__sqrt__ge__one,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xd)) ) ).

% real_sqrt_ge_one
tff(fact_9061_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = code_integer_of_int(zero_zero_int) ).

% zero_integer_def
tff(fact_9062_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,code_integer_of_int(Xaa)),code_integer_of_int(Xd))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xd) ) ).

% less_integer.abs_eq
tff(fact_9063_one__integer__def,axiom,
    one_one_Code_integer = code_integer_of_int(one_one_int) ).

% one_integer_def
tff(fact_9064_real__div__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),aa_real_real(sqrt,Xd)) = aa_real_real(sqrt,Xd) ) ) ).

% real_div_sqrt
tff(fact_9065_sqrt__add__le__add__sqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xd)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9066_le__real__sqrt__sumsq,axiom,
    ! [Xd: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_9067_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% sqrt2_less_2
tff(fact_9068_real__less__rsqrt,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_9069_sqrt__le__D,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) ) ).

% sqrt_le_D
tff(fact_9070_real__le__rsqrt,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_9071_real__le__lsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xd)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_9072_real__sqrt__unique,axiom,
    ! [Y: real,Xd: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))) = Xd )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xd) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_9073_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(bit0(one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9074_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) = Xd )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9075_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) = Y )
     => ( Xd = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9076_real__sqrt__sum__squares__ge1,axiom,
    ! [Xd: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9077_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9078_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(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),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(bit0(one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_9079_sqrt__ge__absD,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd),numeral_numeral_nat(bit0(one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_9080_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% cos_45
tff(fact_9081_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% sin_45
tff(fact_9082_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9083_real__less__lsqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xd)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_9084_sqrt__sum__squares__le__sum,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9085_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))) ) ) ).

% sqrt_even_pow2
tff(fact_9086_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9087_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_ge_abs2
tff(fact_9088_real__sqrt__ge__abs1,axiom,
    ! [Xd: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_ge_abs1
tff(fact_9089_ln__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xd)),numeral_numeral_real(bit0(one))) ) ) ).

% ln_sqrt
tff(fact_9090_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(bit0(one))) ).

% cos_30
tff(fact_9091_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(bit0(one))) ).

% sin_60
tff(fact_9092_arsinh__real__def,axiom,
    ! [Xd: real] : aa_real_real(arsinh_real,Xd) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_9093_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xd)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9094_arsinh__real__aux,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_9095_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_9096_arith__geo__mean__sqrt,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)),numeral_numeral_real(bit0(one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9097_powr__half__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(sqrt,Xd) ) ) ).

% powr_half_sqrt
tff(fact_9098_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9099_cos__x__y__le__one,axiom,
    ! [Xd: 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,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_9100_real__sqrt__sum__squares__less,axiom,
    ! [Xd: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(bit0(one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9101_arcosh__real__def,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
     => ( aa_real_real(arcosh_real,Xd) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9102_cos__arctan,axiom,
    ! [Xd: real] : aa_real_real(cos_real,aa_real_real(arctan,Xd)) = 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,Xd),numeral_numeral_nat(bit0(one)))))) ).

% cos_arctan
tff(fact_9103_sin__arctan,axiom,
    ! [Xd: real] : aa_real_real(sin_real,aa_real_real(arctan,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),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,Xd),numeral_numeral_nat(bit0(one)))))) ).

% sin_arctan
tff(fact_9104_sqrt__sum__squares__half__less,axiom,
    ! [Xd: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(bit0(one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
         => ( 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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9105_sin__cos__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xd))
     => ( aa_real_real(sin_real,Xd) = 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,Xd)),numeral_numeral_nat(bit0(one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9106_arctan__half,axiom,
    ! [Xd: real] : aa_real_real(arctan,Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),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,Xd),numeral_numeral_nat(bit0(one))))))))) ).

% arctan_half
tff(fact_9107_sin__paired,axiom,
    ! [Xd: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jp,Xd)),aa_real_real(sin_real,Xd)) ).

% sin_paired
tff(fact_9108_and__int_Opelims,axiom,
    ! [Xd: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xd),Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xd,Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xd),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xd)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xaa) ))),
                  aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xd)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xd),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,bit0(one)))))) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xd,Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9109_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9110_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_jq),one_one_real) ).

% power_half_series
tff(fact_9111_sums__if_H,axiom,
    ! [G: fun_nat_real,Xd: real] :
      ( aa_real_o(sums_real(G),Xd)
     => aa_real_o(sums_real(aTP_Lamm_jr(G)),Xd) ) ).

% sums_if'
tff(fact_9112_sums__if,axiom,
    ! [G: fun_nat_real,Xd: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xd)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_js(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),Y)) ) ) ).

% sums_if
tff(fact_9113_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,L3: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K2,L3))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,bit0(one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9114_cos__paired,axiom,
    ! [Xd: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jt,Xd)),aa_real_real(cos_real,Xd)) ).

% cos_paired
tff(fact_9115_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,J3: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J3)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(plus_plus_int(I3),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9116_cos__arcsin,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xd)) = 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,Xd),numeral_numeral_nat(bit0(one))))) ) ) ) ).

% cos_arcsin
tff(fact_9117_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_9118_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_9119_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% arcsin_1
tff(fact_9120_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ).

% arcsin_minus_1
tff(fact_9121_arcsin__minus,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xd)) ) ) ) ).

% arcsin_minus
tff(fact_9122_arcsin__le__arcsin,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9123_arcsin__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_9124_arcsin__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd) = aa_real_real(arcsin,Y) )
        <=> ( Xd = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9125_arcsin__less__arcsin,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9126_arcsin__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_9127_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one)))) ).

% set_encode_def
tff(fact_9128_cos__arcsin__nonzero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xd)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9129_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,numeral_numeral_nat(bit0(one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_9130_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),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))),numeral_numeral_nat(bit0(one))) ).

% gauss_sum_nat
tff(fact_9131_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9132_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or4665077453230672383an_nat(M),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,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Ico_nat
tff(fact_9133_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ju(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,numeral_numeral_nat(bit0(one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),numeral_numeral_nat(bit0(one))) ).

% arith_series_nat
tff(fact_9134_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(M),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,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Icc_nat
tff(fact_9135_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9136_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_9137_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ).

% arcsin_ubound
tff(fact_9138_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9139_arcsin__sin,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xd)) = Xd ) ) ) ).

% arcsin_sin
tff(fact_9140_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_9141_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(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_9142_arcsin__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xd)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9143_le__arcsin__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xd))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xd) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9144_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J3)) ) )
     => ( ! [I3: nat,J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J3)),aa_nat_nat(B,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_fr(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9145_Maclaurin__cos__expansion2,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xd)
            & ( aa_real_real(cos_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jv,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9146_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xd: 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,Xd),zero_zero_real)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
            & ( aa_real_real(cos_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jv,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9147_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_9148_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_jw,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
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_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_9151_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_9152_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,numeral_numeral_nat(K)) = insert_nat(pred_numeral(K),aa_nat_set_nat(set_ord_lessThan_nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_9153_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_as,collect_complex(aTP_Lamm_bh(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_9154_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_as,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ag(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_9155_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_9156_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)
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jx(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),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_9157_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jy(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_jz(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_ka(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_9158_Maclaurin__exp__le,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
      & ( aa_real_real(exp_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kb,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_9159_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_aq,set_or1266510415728281911st_int(M,Nb)) = aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_int_int(times_times_int(Nb),aa_int_int(plus_plus_int(Nb),one_one_int))),aa_int_int(times_times_int(M),aa_int_int(minus_minus_int(M),one_one_int)))),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_9160_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_9161_Maclaurin__exp__lt,axiom,
    ! [Xd: real,Nb: nat] :
      ( ( Xd != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
            & ( aa_real_real(exp_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kb,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9162_Maclaurin__sin__expansion,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T5: real] : aa_real_real(sin_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_9163_Maclaurin__sin__expansion2,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
      & ( aa_real_real(sin_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9164_Maclaurin__cos__expansion,axiom,
    ! [Xd: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
      & ( aa_real_real(cos_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jv,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9165_Maclaurin__sin__expansion4,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),Xd)
          & ( aa_real_real(sin_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9166_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xd)
            & ( aa_real_real(sin_real,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc,Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(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,Xd),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9167_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),numeral_numeral_nat(bit0(one))))) ) ) ).

% sin_arccos_abs
tff(fact_9168_sin__arccos,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xd)) = 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,Xd),numeral_numeral_nat(bit0(one))))) ) ) ) ).

% sin_arccos
tff(fact_9169_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9170_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_9171_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_9172_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_9173_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% arccos_0
tff(fact_9174_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_9175_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_9176_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_9177_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,numeral_numeral_nat(K)) = insert_nat(numeral_numeral_nat(K),aa_nat_set_nat(set_ord_atMost_nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_9178_arccos__le__arccos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)) ) ) ) ).

% arccos_le_arccos
tff(fact_9179_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd,M)),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,M)) ).

% sum_choose_upper
tff(fact_9180_arccos__le__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xd) ) ) ) ).

% arccos_le_mono
tff(fact_9181_arccos__eq__iff,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd) = aa_real_real(arccos,Y) )
      <=> ( Xd = Y ) ) ) ).

% arccos_eq_iff
tff(fact_9182_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ke,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_9183_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kf,Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_9184_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kf,Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9185_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_9186_arccos__less__arccos,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd)) ) ) ) ).

% arccos_less_arccos
tff(fact_9187_arccos__less__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd) ) ) ) ).

% arccos_less_mono
tff(fact_9188_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_9189_arccos__cos,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xd)) = Xd ) ) ) ).

% arccos_cos
tff(fact_9190_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_9191_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kg(M),Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9192_vandermonde,axiom,
    ! [M: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_kh(M),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,M),Nb)),R2) ).

% vandermonde
tff(fact_9193_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_9194_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_9195_sin__arccos__nonzero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xd)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9196_arccos__cos2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xd)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xd)) = aa_real_real(uminus_uminus_real,Xd) ) ) ) ).

% arccos_cos2
tff(fact_9197_arccos__minus,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xd)) ) ) ) ).

% arccos_minus
tff(fact_9198_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_9199_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,numeral_numeral_nat(bit0(one))),Nb) ).

% choose_row_sum
tff(fact_9200_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ki(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9201_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_9202_arccos__minus__abs,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xd)) ) ) ).

% arccos_minus_abs
tff(fact_9203_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xd: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J3)
           => ( aa_nat_nat(B,J3) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(A),Xd)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(B),Xd)),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_kl(A),B),Xd)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9204_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_km,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,numeral_numeral_nat(bit0(one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9205_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) ) ) ).

% arccos_le_pi2
tff(fact_9206_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) ).

% binomial_r_part_sum
tff(fact_9207_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn,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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9208_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,numeral_numeral_real(bit0(one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9209_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9210_real__scaleR__def,axiom,
    ! [A: real,Xd: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xd) ).

% real_scaleR_def
tff(fact_9211_Maclaurin__sin__bound,axiom,
    ! [Xd: 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,Xd)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kc,Xd),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,Xd)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9212_real__sqrt__inverse,axiom,
    ! [Xd: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xd)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd)) ).

% real_sqrt_inverse
tff(fact_9213_divide__real__def,axiom,
    ! [Xd: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_9214_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_9215_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_9216_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_9217_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N6: nat] :
          ( ( N6 != 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,N6)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N6))),E) ) ) ).

% real_arch_inverse
tff(fact_9218_sqrt__divide__self__eq,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xd)),Xd) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9219_ln__inverse,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xd)) ) ) ).

% ln_inverse
tff(fact_9220_log__inverse,axiom,
    ! [A: real,Xd: 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),Xd)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xd)) ) ) ) ) ).

% log_inverse
tff(fact_9221_exp__plus__inverse__exp,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xd)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xd)))) ).

% exp_plus_inverse_exp
tff(fact_9222_plus__inverse__ge__2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(inverse_inverse_real,Xd))) ) ).

% plus_inverse_ge_2
tff(fact_9223_real__inv__sqrt__pow2,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd))),numeral_numeral_nat(bit0(one))) = aa_real_real(inverse_inverse_real,Xd) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9224_tan__cot,axiom,
    ! [Xd: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),Xd)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xd)) ).

% tan_cot
tff(fact_9225_real__le__x__sinh,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xd)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xd)))),numeral_numeral_real(bit0(one)))) ) ).

% real_le_x_sinh
tff(fact_9226_real__le__abs__sinh,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),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,Xd)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xd)))),numeral_numeral_real(bit0(one))))) ).

% real_le_abs_sinh
tff(fact_9227_powr__real__of__int,axiom,
    ! [Xd: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),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,Xd),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9228_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T5)),aa_real_real(sin_real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9229_sinh__real__zero__iff,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sinh_real,Xd) = zero_zero_real )
    <=> ( Xd = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9230_sinh__real__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xd)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ).

% sinh_real_less_iff
tff(fact_9231_sinh__real__neg__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9232_sinh__real__pos__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd) ) ).

% sinh_real_pos_iff
tff(fact_9233_sinh__real__nonneg__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xd))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd) ) ).

% sinh_real_nonneg_iff
tff(fact_9234_sinh__real__nonpos__iff,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xd)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9235_norm__cos__sin,axiom,
    ! [T: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T))) = one_one_real ).

% norm_cos_sin
tff(fact_9236_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_9237_cosh__real__nonzero,axiom,
    ! [Xd: real] : aa_real_real(cosh_real,Xd) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9238_complex__of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_real_complex(complex2(R2),zero_zero_real) ).

% complex_of_real_def
tff(fact_9239_complex__of__real__code,axiom,
    ! [X4: real] : aa_real_complex(real_V4546457046886955230omplex,X4) = aa_real_complex(complex2(X4),zero_zero_real) ).

% complex_of_real_code
tff(fact_9240_complex__eq__cancel__iff2,axiom,
    ! [Xd: real,Y: real,Xaa: real] :
      ( ( aa_real_complex(complex2(Xd),Y) = aa_real_complex(real_V4546457046886955230omplex,Xaa) )
    <=> ( ( Xd = Xaa )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_9241_sinh__less__cosh__real,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xd)),aa_real_real(cosh_real,Xd)) ).

% sinh_less_cosh_real
tff(fact_9242_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xd: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xd),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xd)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9243_Complex__mult__complex__of__real,axiom,
    ! [Xd: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(Xd),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9244_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = numeral_numeral_real(W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9245_cosh__real__pos,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xd)) ).

% cosh_real_pos
tff(fact_9246_cosh__real__nonpos__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xd) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9247_cosh__real__nonneg__le__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9248_cosh__real__nonneg,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xd)) ).

% cosh_real_nonneg
tff(fact_9249_cosh__real__ge__1,axiom,
    ! [Xd: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xd)) ).

% cosh_real_ge_1
tff(fact_9250_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_9251_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_9252_cosh__real__strict__mono,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xd)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9253_cosh__real__nonneg__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( 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,Xd)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9254_cosh__real__nonpos__less__iff,axiom,
    ! [Xd: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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,Xd)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xd) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9255_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9256_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_9257_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9258_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9259_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),numeral_numeral_nat(bit0(one)))))) ).

% complex_inverse
tff(fact_9260_arcosh__cosh__real,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xd)) = Xd ) ) ).

% arcosh_cosh_real
tff(fact_9261_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9262_complex__norm,axiom,
    ! [Xd: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xd),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,Xd),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) ).

% complex_norm
tff(fact_9263_cosh__ln__real,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(inverse_inverse_real,Xd))),numeral_numeral_real(bit0(one))) ) ) ).

% cosh_ln_real
tff(fact_9264_sinh__ln__real,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),aa_real_real(inverse_inverse_real,Xd))),numeral_numeral_real(bit0(one))) ) ) ).

% sinh_ln_real
tff(fact_9265_cot__less__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xd)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9266_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9267_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_9268_cot__periodic,axiom,
    ! [Xd: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cot_real,Xd) ).

% cot_periodic
tff(fact_9269_cot__gt__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xd)) ) ) ).

% cot_gt_zero
tff(fact_9270_tan__cot_H,axiom,
    ! [Xd: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),Xd)) = aa_real_real(cot_real,Xd) ).

% tan_cot'
tff(fact_9271_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(bit0(one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9272_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9273_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(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_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_9274_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_9275_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_9276_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9277_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),numeral_numeral_nat(bit0(one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9278_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_9279_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_9280_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),numeral_numeral_nat(bit0(one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9281_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_9282_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_9283_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_9284_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_9285_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_9286_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_9287_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_9288_Complex__eq__i,axiom,
    ! [Xd: real,Y: real] :
      ( ( aa_real_complex(complex2(Xd),Y) = imaginary_unit )
    <=> ( ( Xd = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9289_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9290_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9291_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_9292_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_9293_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,numeral_numeral_nat(bit0(one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_9294_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,numeral_numeral_nat(bit0(one))),Nb)) ).

% mask_nat_less_exp
tff(fact_9295_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,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_9296_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_9297_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,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_9298_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_9299_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,bit0(one))),Nb)),aa_int_int(plus_plus_int(K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9300_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(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_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_9301_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one))) ).

% Arg_minus_ii
tff(fact_9302_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(bit0(one))))) ).

% csqrt_ii
tff(fact_9303_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_9304_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_9305_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9306_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9307_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),numeral_numeral_nat(bit0(one))) = Z ).

% power2_csqrt
tff(fact_9308_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) ).

% Arg_ii
tff(fact_9309_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9310_of__real__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xd)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xd)) ) ) ).

% of_real_sqrt
tff(fact_9311_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9312_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9313_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(bitM(M),bit0(one)) = product_Pair_int_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9314_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_9315_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9316_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = numeral_numeral_nat(bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_9317_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_9318_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9319_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9320_cis__neq__zero,axiom,
    ! [A: real] : aa_real_complex(cis,A) != zero_zero_complex ).

% cis_neq_zero
tff(fact_9321_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_9322_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(bit0(Nb)) = aa_num_num(bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_9323_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa_num_num(bit1,Nb)) = aa_num_num(bit1,bit0(Nb)) ).

% semiring_norm(28)
tff(fact_9324_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = bit0(Nb) ).

% inc_BitM_eq
tff(fact_9325_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_9326_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z)) = aa_complex_complex(sgn_sgn_complex,Z) ) ) ).

% cis_Arg
tff(fact_9327_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : numeral_numeral_nat(bit0(Nb)) = aa_nat_nat(suc,numeral_numeral_nat(bitM(Nb))) ).

% eval_nat_numeral(2)
tff(fact_9328_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bitM(Nb)) = bit0(Nb) ).

% one_plus_BitM
tff(fact_9329_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bitM(Nb)),one) = bit0(Nb) ).

% BitM_plus_one
tff(fact_9330_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_9331_cis__Arg__unique,axiom,
    ! [Z: complex,Xd: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xd) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xd)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),pi)
         => ( aa_complex_real(arg,Z) = Xd ) ) ) ) ).

% cis_Arg_unique
tff(fact_9332_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9333_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_ko(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_bh(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9334_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_kp(L),Qr) ).

% divmod_step_nat_def
tff(fact_9335_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_kq(L),Qr) ).

% divmod_step_int_def
tff(fact_9336_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_kr(L),Qr) ).

% divmod_step_integer_def
tff(fact_9337_divmod__nat__if,axiom,
    ! [M: nat,Nb: nat] :
      divmod_nat(M,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        produc2626176000494625587at_nat(aTP_Lamm_ks,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_9338_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_kt(Z))) ).

% Arg_def
tff(fact_9339_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_ku,Y)) ).

% arctan_def
tff(fact_9340_ln__neg__is__const,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xd) = the_real(aTP_Lamm_kv) ) ) ).

% ln_neg_is_const
tff(fact_9341_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_kw,Y)) ).

% arccos_def
tff(fact_9342_divmod__nat__def,axiom,
    ! [M: nat,Nb: nat] : divmod_nat(M,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% divmod_nat_def
tff(fact_9343_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))) = the_real(aTP_Lamm_kx) ).

% pi_half
tff(fact_9344_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),the_real(aTP_Lamm_kx)) ).

% pi_def
tff(fact_9345_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_ky,Y)) ).

% arcsin_def
tff(fact_9346_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_bh(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_kz(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9347_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_9348_real__root__Suc__0,axiom,
    ! [Xd: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xd) = Xd ).

% real_root_Suc_0
tff(fact_9349_real__root__eq__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xd = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9350_root__0,axiom,
    ! [Xd: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xd) = zero_zero_real ).

% root_0
tff(fact_9351_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd) = zero_zero_real )
      <=> ( Xd = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9352_real__root__less__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),Y) ) ) ).

% real_root_less_iff
tff(fact_9353_real__root__le__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),Y) ) ) ).

% real_root_le_iff
tff(fact_9354_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_9355_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd) = one_one_real )
      <=> ( Xd = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9356_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_9357_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9358_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_9359_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9360_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_9361_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9362_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_9363_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xd: 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),Xd)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9364_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),Nb) = Xd ) ) ) ).

% real_root_pow_pos2
tff(fact_9365_real__root__commute,axiom,
    ! [M: nat,Nb: nat,Xd: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,M),Xd)) ).

% real_root_commute
tff(fact_9366_real__root__mult__exp,axiom,
    ! [M: nat,Nb: nat,Xd: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),Xd) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ).

% real_root_mult_exp
tff(fact_9367_real__root__mult,axiom,
    ! [Nb: nat,Xd: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_mult
tff(fact_9368_real__root__minus,axiom,
    ! [Nb: nat,Xd: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,Xd)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ).

% real_root_minus
tff(fact_9369_real__root__inverse,axiom,
    ! [Nb: nat,Xd: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,Xd)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ).

% real_root_inverse
tff(fact_9370_real__root__divide,axiom,
    ! [Nb: nat,Xd: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_divide
tff(fact_9371_real__root__pos__pos__le,axiom,
    ! [Xd: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => 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),Xd)) ) ).

% real_root_pos_pos_le
tff(fact_9372_real__root__less__mono,axiom,
    ! [Nb: nat,Xd: 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,Xd),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9373_real__root__le__mono,axiom,
    ! [Nb: nat,Xd: 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,Xd),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9374_real__root__power,axiom,
    ! [Nb: nat,Xd: 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,Xd),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),K) ) ) ).

% real_root_power
tff(fact_9375_real__root__abs,axiom,
    ! [Nb: nat,Xd: 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,Xd)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ) ) ).

% real_root_abs
tff(fact_9376_sgn__root,axiom,
    ! [Nb: nat,Xd: 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),Xd)) = aa_real_real(sgn_sgn_real,Xd) ) ) ).

% sgn_root
tff(fact_9377_real__root__gt__zero,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ) ) ).

% real_root_gt_zero
tff(fact_9378_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xd: 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),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9379_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,numeral_numeral_nat(bit0(one))) ).

% sqrt_def
tff(fact_9380_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_9381_real__root__pos__pos,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => 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),Xd)) ) ) ).

% real_root_pos_pos
tff(fact_9382_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xd: 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),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_real_real(aa_nat_fun_real_real(root,N2),Xd)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9383_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xd: 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),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xd)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xd)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)) ) ) ) ).

% real_root_decreasing
tff(fact_9384_odd__real__root__pow,axiom,
    ! [Nb: nat,Xd: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),Nb) = Xd ) ) ).

% odd_real_root_pow
tff(fact_9385_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xd: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xd )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xd) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9386_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xd: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = Xd ) ) ).

% odd_real_root_power_cancel
tff(fact_9387_real__root__pow__pos,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),Nb) = Xd ) ) ) ).

% real_root_pow_pos
tff(fact_9388_real__root__power__cancel,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb)) = Xd ) ) ) ).

% real_root_power_cancel
tff(fact_9389_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xd: 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) = Xd )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xd) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9390_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xd: 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),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),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),Xd)),aa_real_real(aa_nat_fun_real_real(root,N2),Xd)) ) ) ) ) ).

% real_root_increasing
tff(fact_9391_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_9392_sgn__power__root,axiom,
    ! [Nb: nat,Xd: 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),Xd))),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),Xd))),Nb)) = Xd ) ) ).

% sgn_power_root
tff(fact_9393_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9394_log__base__root,axiom,
    ! [Nb: nat,B: real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xd) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xd)) ) ) ) ).

% log_base_root
tff(fact_9395_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9396_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xd: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd))
    <=> ( ( ( 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)) = Xd )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9397_root__powr__inverse,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xd) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xd),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9398_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9399_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_9400_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_9401_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9402_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_9403_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_int_fun_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_9404_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9405_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),numeral_numeral_nat(Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9406_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),numeral_numeral_nat(Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9407_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,bit0(W))),Nb)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9408_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),Nb)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9409_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_9410_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),Nb)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(minus_minus_int(K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9411_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_9412_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9413_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_9414_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_9415_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(divide_divide_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% not_int_div_2
tff(fact_9416_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,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,bit0(one))),K) ) ).

% even_not_iff_int
tff(fact_9417_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(4)
tff(fact_9418_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,bit0(Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9419_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_int_fun_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(M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9420_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M3)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_9421_and__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9422_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(7)
tff(fact_9423_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_9424_and__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9425_and__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9426_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) ) ).

% bit_int_def
tff(fact_9427_and__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9428_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% not_int_rec
tff(fact_9429_Bit__Operations_Oset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(plus_plus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) ).

% Bit_Operations.set_bit_eq
tff(fact_9430_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb))) ).

% unset_bit_eq
tff(fact_9431_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_9432_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9433_bitNOT__integer__code,axiom,
    ! [I: code_integer] : bit_ri7632146776885996613nteger(I) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(I),one_one_Code_integer) ).

% bitNOT_integer_code
tff(fact_9434_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9435_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_9436_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_9437_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_9438_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_9439_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_9440_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_9441_or__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,Xd)) ).

% or_nat_numerals(4)
tff(fact_9442_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)),numeral_numeral_nat(aa_num_num(bit1,Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9443_or__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit0(Xd))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,Xd)) ).

% or_nat_numerals(3)
tff(fact_9444_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)),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9445_or__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9446_and__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9447_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_9448_int__or__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se1409905431419307370or_int(I),zero_zero_int) = I ).

% int_or_code(2)
tff(fact_9449_int__or__code_I1_J,axiom,
    ! [J: int] : aa_int_int(bit_se1409905431419307370or_int(zero_zero_int),J) = J ).

% int_or_code(1)
tff(fact_9450_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),Nb)
    <=> ~ ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9451_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9452_or__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9453_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_9454_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_9455_int__xor__code_I1_J,axiom,
    ! [J: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),J) = J ).

% int_xor_code(1)
tff(fact_9456_int__xor__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se6526347334894502574or_int(I),zero_zero_int) = I ).

% int_xor_code(2)
tff(fact_9457_XOR__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( 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(Xd),Y)) ) ) ).

% XOR_lower
tff(fact_9458_OR__lower,axiom,
    ! [Xd: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( 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(Xd),Y)) ) ) ).

% OR_lower
tff(fact_9459_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_9460_plus__and__or,axiom,
    ! [Xd: int,Y: int] : aa_int_int(plus_plus_int(aa_int_int(bit_se725231765392027082nd_int(Xd),Y)),aa_int_int(bit_se1409905431419307370or_int(Xd),Y)) = aa_int_int(plus_plus_int(Xd),Y) ).

% plus_and_or
tff(fact_9461_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_9462_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)),numeral_numeral_nat(Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_9463_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_9464_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9465_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,bit0(Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(Nb))) ).

% or_not_numerals(2)
tff(fact_9466_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_9467_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,bit0(Nb))) ).

% or_not_numerals(3)
tff(fact_9468_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9469_bit__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb))) ) ).

% bit_nat_def
tff(fact_9470_or__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9471_XOR__upper,axiom,
    ! [Xd: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,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,bit0(one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xd),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9472_OR__upper,axiom,
    ! [Xd: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xd)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xd),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,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,bit0(one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xd),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9473_or__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9474_test__bit__int__code_I1_J,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,zero_zero_int),Nb) ).

% test_bit_int_code(1)
tff(fact_9475_int__and__code_I2_J,axiom,
    ! [I: int] : aa_int_int(bit_se725231765392027082nd_int(I),zero_zero_int) = zero_zero_int ).

% int_and_code(2)
tff(fact_9476_int__and__code_I1_J,axiom,
    ! [J: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),J) = zero_zero_int ).

% int_and_code(1)
tff(fact_9477_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9478_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9479_or__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))) ).

% or_nat_rec
tff(fact_9480_or__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9481_or__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9482_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% xor_int_rec
tff(fact_9483_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% or_int_rec
tff(fact_9484_or__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))))))) ) ).

% or_nat_unfold
tff(fact_9485_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,bit0(one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ).

% or_int_unfold
tff(fact_9486_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xd: $o] : bits_Bit_integer(code_integer_of_int(Xaa),(Xd)) = code_integer_of_int(aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(Xd))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Xaa))) ).

% Bit_integer.abs_eq
tff(fact_9487_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,bit0(Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(Nb)))) ).

% or_minus_numerals(1)
tff(fact_9488_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,bit0(Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(Nb)))) ).

% or_minus_numerals(5)
tff(fact_9489_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)),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9490_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)),numeral_numeral_nat(aa_num_num(bit1,Y))) = numeral_numeral_nat(bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_9491_xor__nat__numerals_I3_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit0(Xd))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(aa_num_num(bit1,Xd)) ).

% xor_nat_numerals(3)
tff(fact_9492_xor__nat__numerals_I4_J,axiom,
    ! [Xd: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(aa_num_num(bit1,Xd))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit0(Xd)) ).

% xor_nat_numerals(4)
tff(fact_9493_or__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bit0(Nb)))) ).

% or_minus_numerals(4)
tff(fact_9494_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,M: 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,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bit0(Nb)))) ).

% or_minus_numerals(8)
tff(fact_9495_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Nb)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(Nb)))) ).

% or_minus_numerals(7)
tff(fact_9496_or__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(Nb)))) ).

% or_minus_numerals(3)
tff(fact_9497_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9498_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(Nb)),one) = bit0(one) ).

% or_not_num_neg.simps(4)
tff(fact_9499_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(Nb)),aa_num_num(bit1,M)) = bit0(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9500_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9501_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9502_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,bit0(Nb)),bit0(M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9503_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit1,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9504_xor__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9505_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bit0(M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9506_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),bit0(M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9507_or__not__num__neg_Oelims,axiom,
    ! [Xd: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xd),Xaa) = Y )
     => ( ( ( Xd = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xd = one )
           => ! [M2: num] :
                ( ( Xaa = bit0(M2) )
               => ( Y != aa_num_num(bit1,M2) ) ) )
         => ( ( ( Xd = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit1,M2) )
                 => ( Y != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N: num] : Xd = bit0(N)
               => ( ( Xaa = one )
                 => ( Y != bit0(one) ) ) )
             => ( ! [N: num] :
                    ( ( Xd = bit0(N) )
                   => ! [M2: num] :
                        ( ( Xaa = bit0(M2) )
                       => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xd = bit0(N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit1,M2) )
                         => ( Y != bit0(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                 => ( ( ? [N: num] : Xd = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xd = aa_num_num(bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = bit0(M2) )
                             => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xd = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit1,M2) )
                               => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9508_numeral__or__not__num__eq,axiom,
    ! [M: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9509_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9510_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9511_xor__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one))))))) ) ).

% xor_nat_unfold
tff(fact_9512_xor__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),numeral_numeral_nat(bit0(one)))))) ).

% xor_nat_rec
tff(fact_9513_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9514_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9515_int__lsb__numeral_I1_J,axiom,
    ~ aa_int_o(least_4859182151741483524sb_int,zero_zero_int) ).

% int_lsb_numeral(1)
tff(fact_9516_int__lsb__numeral_I2_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,one_one_int) ).

% int_lsb_numeral(2)
tff(fact_9517_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,bit0(W))) ).

% int_lsb_numeral(6)
tff(fact_9518_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_9519_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_9520_int__lsb__numeral_I4_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% int_lsb_numeral(4)
tff(fact_9521_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))) ).

% int_lsb_numeral(8)
tff(fact_9522_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_9523_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_9524_lsb__integer__code,axiom,
    ! [Xd: code_integer] :
      ( aa_Code_integer_o(least_7544222001954398261nteger,Xd)
    <=> aa_nat_o(bit_se9216721137139052372nteger(Xd),zero_zero_nat) ) ).

% lsb_integer_code
tff(fact_9525_lsb__int__def,axiom,
    ! [I: int] :
      ( aa_int_o(least_4859182151741483524sb_int,I)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,I),zero_zero_nat) ) ).

% lsb_int_def
tff(fact_9526_bin__last__conv__lsb,axiom,
    aTP_Lamm_la = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9527_vebt__buildup_Opelims,axiom,
    ! [Xd: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xd) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xd)
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xd = 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] :
                  ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
                            aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))),
                            aa_VEB5832953974749077948T_VEBT(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,numeral_numeral_nat(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_9528_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xd) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xd)
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y = numeral_numeral_nat(bit0(bit0(one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = numeral_numeral_nat(bit0(bit0(one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,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))),numeral_numeral_nat(bit0(one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_9529_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xd) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xd)
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y = numeral_numeral_nat(aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = numeral_numeral_nat(aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_9530_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xd: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xd) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xd)
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xd = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xd = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(bit0(one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9531_dup__1,axiom,
    code_dup(one_one_Code_integer) = numera6620942414471956472nteger(bit0(one)) ).

% dup_1
tff(fact_9532_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% Code_Numeral.dup_code(1)
tff(fact_9533_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(pure_assn(( ( Mmoi = Mmo )
                & ( Degi = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_lb(Tree_list),Tree_array))) ).

% vebt_assn_raw.simps(2)
tff(fact_9534_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9535_sin__times__pi__eq__0,axiom,
    ! [Xd: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xd),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xd),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9536_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9537_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9538_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = numeral_numeral_nat(aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = 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(plus_plus_nat,numeral_numeral_nat(aa_num_num(bit1,bit0(one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9539_merge__true__star,axiom,
    aa_assn_assn(times_times_assn(top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_9540_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_9541_ent__true,axiom,
    ! [P: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,P),top_top_assn) ).

% ent_true
tff(fact_9542_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(top_top_assn),aa_assn_assn(times_times_assn(top_top_assn),P)) = aa_assn_assn(times_times_assn(top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_9543_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),Q)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_9544_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(P),R)),aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_9545_ent__refl__true,axiom,
    ! [A2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(times_times_assn(A2),top_top_assn)) ).

% ent_refl_true
tff(fact_9546_ent__star__mono__true,axiom,
    ! [A2: assn,A5: assn,B3: assn,B9: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(times_times_assn(A5),top_top_assn))
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B3),aa_assn_assn(times_times_assn(B9),top_top_assn))
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B3)),top_top_assn)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A5),B9)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_9547_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H) ) ).

% mod_star_trueI
tff(fact_9548_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H5) ) ).

% mod_star_trueE
tff(fact_9549_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_9550_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = numeral_numeral_nat(bit0(bit0(one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9551_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = one_one_real )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),vEBT_Leaf((A4),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_9552_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B2))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9553_vebt__maxt_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      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),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9554_vebt__mint_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = $ite(
                      (A4),
                      some_nat(zero_zero_nat),
                      $ite((B2),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9555_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A4),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9556_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf((A4),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) ) )
           => ~ ! [Mi: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9557_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one_nat )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xd = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xd = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9558_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xd)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xd = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xd = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9559_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xd)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xd)
       => ( ( ( Xd = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9560_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( modulo364778990260209775nteger(I,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9561_bin__rest__integer_Oabs__eq,axiom,
    ! [Xd: int] : bits_b2549910563261871055nteger(code_integer_of_int(Xd)) = code_integer_of_int(aa_int_int(divide_divide_int(Xd),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9562_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : zero_n356916108424825756nteger(aa_Code_integer_o(bits_b8758750999018896077nteger,I)) = bit_se3949692690581998587nteger(I,one_one_Code_integer) ).

% bitval_bin_last_integer
tff(fact_9563_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( bit_se3949692690581998587nteger(I,one_one_Code_integer) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_code
tff(fact_9564_bitAND__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      bit_se3949692690581998587nteger(Xd,Y) = $ite(
        Xd = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(
          Xd = uminus1351360451143612070nteger(one_one_Code_integer),
          Y,
          bits_Bit_integer(bit_se3949692690581998587nteger(bits_b2549910563261871055nteger(Xd),bits_b2549910563261871055nteger(Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xd)
            & aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_9565_bitOR__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      bit_se1080825931792720795nteger(Xd,Y) = $ite(
        Xd = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xd = uminus1351360451143612070nteger(one_one_Code_integer),
          uminus1351360451143612070nteger(one_one_Code_integer),
          bits_Bit_integer(bit_se1080825931792720795nteger(bits_b2549910563261871055nteger(Xd),bits_b2549910563261871055nteger(Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xd)
            | aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_9566_bitXOR__integer__unfold,axiom,
    ! [Xd: code_integer,Y: code_integer] :
      bit_se3222712562003087583nteger(Xd,Y) = $ite(
        Xd = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xd = uminus1351360451143612070nteger(one_one_Code_integer),
          bit_ri7632146776885996613nteger(Y),
          bits_Bit_integer(bit_se3222712562003087583nteger(bits_b2549910563261871055nteger(Xd),bits_b2549910563261871055nteger(Y)),
            ~ aa_Code_integer_o(bits_b8758750999018896077nteger,Xd) = aa_Code_integer_o(bits_b8758750999018896077nteger,Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_9567_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = divide6298287555418463151nteger(I,numera6620942414471956472nteger(bit0(one))) ).

% bin_rest_integer_code
tff(fact_9568_bin__last__integer_Oabs__eq,axiom,
    ! [Xd: int] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,code_integer_of_int(Xd))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xd) ) ).

% bin_last_integer.abs_eq
tff(fact_9569_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(minus_minus_int(J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9570_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(bit_concat_bit(Nb,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_9571_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_9572_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_9573_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_9574_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_9575_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_9576_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_9577_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_9578_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,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_9579_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9580_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = aa_int_int(bit_se8568078237143864401it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9581_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,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9582_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(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(pred_numeral(L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9583_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_9584_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se8568078237143864401it_int(zero_zero_nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_9585_drop__bit__int__code_I2_J,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9586_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,K),aa_int_int(bit_concat_bit(M,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R2) ).

% concat_bit_assoc
tff(fact_9587_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : bit_concat_bit(Nb,aa_int_int(bit_se2923211474154528505it_int(Nb),B)) = bit_concat_bit(Nb,B) ).

% concat_bit_take_bit_eq
tff(fact_9588_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_9589_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9590_shiftr__integer__conv__div__pow2,axiom,
    ! [Nb: nat,Xd: code_integer] : bit_se3928097537394005634nteger(Nb,Xd) = divide6298287555418463151nteger(Xd,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9591_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(divide_divide_int(I),aa_num_int(numeral_numeral_int,bit0(one))) = aa_int_int(bit_se8568078237143864401it_int(one_one_nat),I) ).

% bin_rest_code
tff(fact_9592_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K) = aa_int_int(divide_divide_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% drop_bit_int_def
tff(fact_9593_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9594_drop__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9595_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_9596_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_9597_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_9598_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,numeral_numeral_nat(bit0(one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9599_drop__bit__push__bit__int,axiom,
    ! [M: nat,Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),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,M),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9600_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se545348938243370406it_int(zero_zero_nat),I) = I ).

% push_bit_int_code(1)
tff(fact_9601_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_9602_set__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9603_flip__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9604_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = bit_se7788150548672797655nteger(one_one_nat,I) ).

% Bit_integer_code(1)
tff(fact_9605_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9606_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9607_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_9608_Bit__Operations_Oset__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)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9609_bit__push__bit__iff__nat,axiom,
    ! [M: 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,M),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),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),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9610_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_9611_shiftl__integer__conv__mult__pow2,axiom,
    ! [Nb: nat,Xd: code_integer] : bit_se7788150548672797655nteger(Nb,Xd) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Xd),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one))),Nb)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9612_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_9613_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K) = aa_int_int(times_times_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Nb)) ).

% push_bit_int_def
tff(fact_9614_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = plus_p5714425477246183910nteger(bit_se7788150548672797655nteger(one_one_nat,I),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_9615_push__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) ).

% push_bit_nat_def
tff(fact_9616_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,bit0(one))),Nb)) ).

% push_bit_minus_one
tff(fact_9617_set__bit__integer__conv__masks,axiom,
    ! [Xd: code_integer,I: nat,B: $o] :
      generi2397576812484419408nteger(Xd,I,(B)) = $ite((B),bit_se1080825931792720795nteger(Xd,bit_se7788150548672797655nteger(I,one_one_Code_integer)),bit_se3949692690581998587nteger(Xd,bit_ri7632146776885996613nteger(bit_se7788150548672797655nteger(I,one_one_Code_integer)))) ).

% set_bit_integer_conv_masks
tff(fact_9618_Uint32__code,axiom,
    ! [I: code_integer] :
      aa_Cod8401701657498852871uint32(uint322,I) = $let(
        i: code_integer,
        i:= bit_se3949692690581998587nteger(I,numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))))),
        $ite(aa_nat_o(bit_se9216721137139052372nteger(i),numeral_numeral_nat(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),uint32_signed(minus_8373710615458151222nteger(i,numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))))))))))))))))))))))))))),uint32_signed(i)) ) ).

% Uint32_code
tff(fact_9619_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      uint32_signed(I) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),uminus1351360451143612070nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))))))))))))))))))))))))))
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))))))))))))))))))))))))),I) ),
        aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(undefi1969809534184926113uint32,uint322),I),
        aa_Cod8401701657498852871uint32(uint322,I) ) ).

% Uint32_signed_def
tff(fact_9620_Generic__set__bit_Oset__bit__int__def,axiom,
    ! [I: int,Nb: nat,B: $o] :
      generi8991105624351003935it_int(I,Nb,(B)) = aa_int_int(aa_nat_fun_int_int($ite((B),bit_se7879613467334960850it_int,bit_se4203085406695923979it_int),Nb),I) ).

% Generic_set_bit.set_bit_int_def
tff(fact_9621_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Nb: nat] : generi8991105624351003935it_int(I,Nb,$true) = aa_int_int(bit_se1409905431419307370or_int(I),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% int_set_bit_True_conv_OR
tff(fact_9622_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Nb: nat] : generi8991105624351003935it_int(I,Nb,$false) = aa_int_int(bit_se725231765392027082nd_int(I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% int_set_bit_False_conv_NAND
tff(fact_9623_int__set__bit__conv__ops,axiom,
    ! [I: int,Nb: nat,B: $o] :
      generi8991105624351003935it_int(I,Nb,(B)) = $ite((B),aa_int_int(bit_se1409905431419307370or_int(I),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)),aa_int_int(bit_se725231765392027082nd_int(I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)))) ).

% int_set_bit_conv_ops
tff(fact_9624_assn__aci_I11_J,axiom,
    ! [Xd: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(syntax7398250324933576852n_assn,aa_assn_assn(times_times_assn(Xd),Y)),A)
     => ( aa_assn_assn(times_times_assn(A),B) = aa_assn_assn(times_times_assn(B),A) ) ) ).

% assn_aci(11)
tff(fact_9625_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,bit0(one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9626_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J2: nat] :
        ? [M8: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M4)
         => ! [N6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N6)
             => 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(X7,M4)),aa_nat_real(X7,N6)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J2)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9627_int__sdiv__simps_I2_J,axiom,
    ! [A: int] : aa_int_int(signed6714573509424544716de_int(A),zero_zero_int) = zero_zero_int ).

% int_sdiv_simps(2)
tff(fact_9628_sdiv__int__0__div,axiom,
    ! [Xd: int] : aa_int_int(signed6714573509424544716de_int(zero_zero_int),Xd) = zero_zero_int ).

% sdiv_int_0_div
tff(fact_9629_sdiv__int__div__0,axiom,
    ! [Xd: int] : aa_int_int(signed6714573509424544716de_int(Xd),zero_zero_int) = zero_zero_int ).

% sdiv_int_div_0
tff(fact_9630_int__sdiv__simps_I1_J,axiom,
    ! [A: int] : aa_int_int(signed6714573509424544716de_int(A),one_one_int) = A ).

% int_sdiv_simps(1)
tff(fact_9631_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(signed6714573509424544716de_int(A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_9632_int__sdiv__simps_I3_J,axiom,
    ! [A: int] : aa_int_int(signed6714573509424544716de_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% int_sdiv_simps(3)
tff(fact_9633_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(signed6714573509424544716de_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% sdiv_int_numeral_numeral
tff(fact_9634_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_9635_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( aa_int_int(signed6714573509424544716de_int(A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_9636_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_9637_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_9638_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M,cons_nat(Nb,Ns)) = upt(M,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9639_map__Suc__upt,axiom,
    ! [M: nat,Nb: nat] : map_nat_nat(suc,upt(M,Nb)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9640_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lc,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_9641_map__add__upt,axiom,
    ! [Nb: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lc,Nb),upt(zero_zero_nat,M)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% map_add_upt
tff(fact_9642_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_9643_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xd: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xd,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xd )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9644_map__decr__upt,axiom,
    ! [M: nat,Nb: nat] : map_nat_nat(aTP_Lamm_ld,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Nb))) = upt(M,Nb) ).

% map_decr_upt
tff(fact_9645_atLeastAtMost__upt,axiom,
    ! [Nb: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),M) = set_nat2(upt(Nb,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9646_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_9647_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(signed6714573509424544716de_int(A),B) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(signed6714573509424544716de_int(A),B)) = aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A),B)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_9648_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_9649_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(bit_se2923211474154528505it_int(Nb),L) )
     => ( map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,Nb)) = map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,Nb)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_9650_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(signed6714573509424544716de_int(K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9651_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),top_top_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% entails_solve_init(1)
tff(fact_9652_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,aa_VEB5832953974749077948T_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_9653_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),F2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),F2)) ) ).

% frame_inference_init
tff(fact_9654_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),F2)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,P),aa_assn_assn(times_times_assn(Q),F2)) ) ).

% FI_QUERY_def
tff(fact_9655_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_QUERY(P),Q),one_one_assn)
     => aa_assn_o(aa_assn_fun_assn_o(entails,P),Q) ) ).

% entails_solve_init(2)
tff(fact_9656_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,aa_VEB5832953974749077948T_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_9657_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => aa_set_int_o(member_int(aa_int_int(signed6292675348222524329lo_int(A),B)),set_or1266510415728281911st_int(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B))),one_one_int),aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,B)),one_one_int))) ) ).

% smod_int_range
tff(fact_9658_smod__int__0__mod,axiom,
    ! [Xd: int] : aa_int_int(signed6292675348222524329lo_int(zero_zero_int),Xd) = zero_zero_int ).

% smod_int_0_mod
tff(fact_9659_smod__int__mod__0,axiom,
    ! [Xd: int] : aa_int_int(signed6292675348222524329lo_int(Xd),zero_zero_int) = Xd ).

% smod_int_mod_0
tff(fact_9660_smod__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(signed6292675348222524329lo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% smod_int_numeral_numeral
tff(fact_9661_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),B) ) ) ).

% smod_int_compares(1)
tff(fact_9662_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(2)
tff(fact_9663_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_9664_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(6)
tff(fact_9665_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_9666_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(8)
tff(fact_9667_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_int(signed6292675348222524329lo_int(A),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% smod_mod_positive
tff(fact_9668_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(3)
tff(fact_9669_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),aa_int_int(uminus_uminus_int,B)) ) ) ).

% smod_int_compares(5)
tff(fact_9670_smod__int__alt__def,axiom,
    ! [A: int,B: int] : aa_int_int(signed6292675348222524329lo_int(A),B) = aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,A)),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% smod_int_alt_def
tff(fact_9671_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9672_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D) ) ).

% valid_eq2
tff(fact_9673_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D) ) ).

% valid_eq
tff(fact_9674_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D) ) ).

% valid_eq1
tff(fact_9675_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9676_uint32_Osize__eq__length,axiom,
    numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9677_len__num0,axiom,
    ! [Uu: itself_Numeral_num0] : type_l4264026598287037464l_num0(Uu) = zero_zero_nat ).

% len_num0
tff(fact_9678_len__num1,axiom,
    ! [Uu: itself_Numeral_num1] : type_l4264026598287037465l_num1(Uu) = one_one_nat ).

% len_num1
tff(fact_9679_len__of__finite__1__def,axiom,
    ! [Xd: itself_finite_1] : type_l31302759751748491nite_1(Xd) = one_one_nat ).

% len_of_finite_1_def
tff(fact_9680_len__of__finite__3__def,axiom,
    ! [Xd: itself_finite_3] : type_l31302759751748493nite_3(Xd) = numeral_numeral_nat(bit0(bit0(one))) ).

% len_of_finite_3_def
tff(fact_9681_len__of__finite__2__def,axiom,
    ! [Xd: itself_finite_2] : type_l31302759751748492nite_2(Xd) = numeral_numeral_nat(bit0(one)) ).

% len_of_finite_2_def
tff(fact_9682_bij__betw__Suc,axiom,
    ! [M7: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M7),N2)
    <=> ( image_nat_nat(suc,M7) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9683_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_9684_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_9685_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9686_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_9687_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_9688_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_9689_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_9690_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_9691_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_9692_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_le(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(minus_minus_int(U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9693_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_lf(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,minus_8373710615458151222nteger(U,L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9694_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_9695_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xd: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lg,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xd),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,Xd),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,Xd),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9696_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,S),M)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Listy))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Nb) )
       => ( ( M = aa_nat_nat(suc,Nb) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9697_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Nb: nat,X14: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => 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_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ) ).

% max_idx_list
tff(fact_9698_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),set_VEBT_VEBT2(TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ) ).

% height_compose_list
tff(fact_9699_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : 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_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(lattic8265883725875713057ax_nat,insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))) ).

% max_ins_scaled
tff(fact_9700_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))) ) ).

% height_i_max
tff(fact_9701_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_9702_divide__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_lh(M),Nb)))) ).

% divide_nat_def
tff(fact_9703_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9704_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xd) = Y )
     => ( ( ? [A4: $o,B2: $o] : Xd = vEBT_Leaf((A4),(B2))
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9705_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xd) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xd)
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ( ( Y = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B2))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9706_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_al,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_9707_min__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)) ).

% min_Suc_Suc
tff(fact_9708_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_9709_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_9710_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9711_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9712_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_9713_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_9714_min__pm,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% min_pm
tff(fact_9715_min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A),B)) = A ).

% min_pm1
tff(fact_9716_rev__min__pm,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ).

% rev_min_pm
tff(fact_9717_rev__min__pm1,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),A)) = A ).

% rev_min_pm1
tff(fact_9718_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_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,pred_numeral(K)),Nb)) ).

% min_numeral_Suc
tff(fact_9719_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_9720_concat__bit__assoc__sym,axiom,
    ! [M: nat,Nb: nat,K: int,L: int,R2: int] : aa_int_int(bit_concat_bit(M,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,M),Nb),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9721_nat__mult__min__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),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,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_9722_nat__mult__min__left,axiom,
    ! [M: 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,M),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_min_left
tff(fact_9723_min__diff,axiom,
    ! [M: 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,M),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,M),Nb)),I) ).

% min_diff
tff(fact_9724_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),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,M),Nb),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9725_mod__mod__power,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb))) ).

% mod_mod_power
tff(fact_9726_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(Q2,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9727_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9728_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_9729_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_li,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9730_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_9731_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_lj) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_9732_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_lk) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_9733_root__def,axiom,
    ! [Nb: nat,Xd: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xd) = $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_ll,Nb)),Xd)) ).

% root_def
tff(fact_9734_bin__last__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),bit_bi6516823479961619367ts_int(F))
      <=> aa_nat_o(F,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_9735_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_lm(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_9736_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(F,zero_zero_nat))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_9737_bin__rest__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(divide_divide_int(bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,bit0(one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_9738_shiftl__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Nb) ).

% shiftl_Suc_0
tff(fact_9739_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_9740_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9741_shiftr__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% shiftr_Suc_0
tff(fact_9742_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xd: real,D3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xd != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
             => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),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,numeral_numeral_nat(bit0(one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
               => ( D3 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),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,numeral_numeral_nat(bit0(one))),Nb)
               => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D3,topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9743_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xd: 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,numeral_numeral_nat(bit0(one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9744_inj__on__diff__nat,axiom,
    ! [N2: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lg,K)),N2) ) ).

% inj_on_diff_nat
tff(fact_9745_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9746_inj__Suc,axiom,
    ! [N2: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N2) ).

% inj_Suc
tff(fact_9747_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9748_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F4,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9749_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xd: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,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)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xd)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9750_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xd: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,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)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),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,Xd),H6))),aa_real_real(F,Xd)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9751_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9752_DERIV__ln,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xd),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9753_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9754_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9755_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9756_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9757_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9758_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xd: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: 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,Xd),Y2))),D)
             => ( aa_real_real(F,Xd) = aa_real_real(F,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9759_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xd: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,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)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),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,Xd),H6))),aa_real_real(F,Xd)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9760_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xd: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,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)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xd)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9761_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,Xd: real,Y: real] :
      ( ! [X: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_real(F,Xd) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9762_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xd: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),H6)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),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,Xd),H6))),aa_real_real(F,Xd)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9763_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xd: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),H6)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xd)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xd),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9764_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xd: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),H6)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),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,Xd),H6))),aa_real_real(F,Xd)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9765_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xd: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),H6)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xd)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9766_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(bit0(one))) ) ) ) ).

% DERIV_const_average
tff(fact_9767_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xd: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: 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,Xd),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xd)),aa_real_real(F,Y2)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9768_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xd: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: 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,Xd),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,Xd)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9769_DERIV__ln__divide,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xd),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9770_DERIV__pow,axiom,
    ! [Nb: nat,Xd: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ln,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,Xd),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xd,S)) ).

% DERIV_pow
tff(fact_9771_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xd: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_lo(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,Xd)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),M),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9772_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_lp,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9773_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9774_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xd: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xd))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_lq(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,Xd)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9775_DERIV__log,axiom,
    ! [Xd: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),Xd)),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9776_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xd: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xd))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xd,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_lr(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xd)),aa_real_real(F,Xd))),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,Xd)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,Xd))),aa_real_real(G,Xd)))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9777_DERIV__real__sqrt,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd))),numeral_numeral_real(bit0(one))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9778_DERIV__arctan,axiom,
    ! [Xd: 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,Xd),numeral_numeral_nat(bit0(one))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9779_arsinh__real__has__field__derivative,axiom,
    ! [Xd: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),numeral_numeral_nat(bit0(one)))),one_one_real))),topolo2177554685111907308n_real(Xd,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_9780_DERIV__real__sqrt__generic,axiom,
    ! [Xd: real,D3: real] :
      ( ( Xd != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xd)
         => ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd))),numeral_numeral_real(bit0(one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),zero_zero_real)
           => ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xd)))),numeral_numeral_real(bit0(one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9781_arcosh__real__has__field__derivative,axiom,
    ! [Xd: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => 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,Xd),numeral_numeral_nat(bit0(one)))),one_one_real))),topolo2177554685111907308n_real(Xd,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9782_artanh__real__has__field__derivative,axiom,
    ! [Xd: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),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,Xd),numeral_numeral_nat(bit0(one))))),topolo2177554685111907308n_real(Xd,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9783_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => 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_9784_DERIV__real__root,axiom,
    ! [Nb: nat,Xd: 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),Xd)
       => 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),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9785_DERIV__arccos,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd),numeral_numeral_nat(bit0(one))))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9786_DERIV__arcsin,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),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,Xd),numeral_numeral_nat(bit0(one)))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9787_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => ( ! [X: nat] :
                ( ~ aa_set_nat_o(member_nat(X),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9788_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xd: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X),topolo2177554685111907308n_real(X,top_top_set_real))
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
            & ( aa_real_real(F,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ls(Diff),Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9789_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xd: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
          & ( aa_real_real(F,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ls(Diff),Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9790_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xd: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
     => ( ( Xd != 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),Xd)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9791_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lt(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9792_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lt(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9793_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lt(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9794_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xd: 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)
       => ( ( Xd != zero_zero_real )
         => ( ! [M2: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X),topolo2177554685111907308n_real(X,top_top_set_real))
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
                & ( aa_real_real(F,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ls(Diff),Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9795_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xd: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xd))
            & ( aa_real_real(F,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ls(Diff),Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xd),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9796_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lu(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9797_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lu(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9798_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xd: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xd)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),B)
                 => ( ( Xd != C )
                   => ? [T5: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xd) ) )
                        & ( aa_real_real(F,Xd) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lv(Diff),C),Xd),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xd),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9799_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_ll,Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9800_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B3: real] :
      ( ! [M2: nat,T5: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M3: nat,T6: 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),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_lx(Nb),Diff),B3),M3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ly(Diff),M3),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3))))),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,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M3))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9801_DERIV__arctan__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_lz,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ma,Xd)),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9802_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_mb(F),X)) )
     => ( aa_set_real_o(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_md(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_mb(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9803_tanh__real__bounds,axiom,
    ! [Xd: real] : aa_set_real_o(member_real(aa_real_real(tanh_real,Xd)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9804_DERIV__isconst3,axiom,
    ! [A: real,B: real,Xd: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_set_real_o(member_real(Xd),set_or1633881224788618240n_real(A,B))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) )
           => ( aa_real_real(F,Xd) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9805_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_me(F),N),aa_nat_real(aa_real_fun_nat_real(F4,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X: real,Y2: real] :
                    ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
                   => ( aa_set_real_o(member_real(Y2),set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X),N)),aa_nat_real(aa_real_fun_nat_real(F,Y2),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_mf(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9806_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_9807_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_9808_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_9809_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_9810_isCont__real__sqrt,axiom,
    ! [Xd: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9811_isCont__real__root,axiom,
    ! [Xd: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_9812_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9813_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),M) = set_nat2(upt(aa_nat_nat(suc,Nb),M)) ).

% greaterThanLessThan_upt
tff(fact_9814_isCont__inverse__function2,axiom,
    ! [A: real,Xd: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),B)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xd),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9815_isCont__ln,axiom,
    ! [Xd: real] :
      ( ( Xd != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_9816_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9817_isCont__arcosh,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9818_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_mg,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9819_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D3: real,G: fun_real_real,Xd: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D3,topolo2177554685111907308n_real(aa_real_real(G,Xd),top_top_set_real))
     => ( ( D3 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xd)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),B)
           => ( ! [Y2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B)
                   => ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D3),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9820_isCont__arccos,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9821_isCont__arcsin,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9822_LIM__less__bound,axiom,
    ! [B: real,Xd: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xd)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(B,Xd))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xd)) ) ) ) ).

% LIM_less_bound
tff(fact_9823_isCont__artanh,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xd,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9824_isCont__inverse__function,axiom,
    ! [D: real,Xd: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z2: 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,Z2),Xd))),D)
           => ( aa_real_real(G,aa_real_real(F,Z2)) = Z2 ) )
       => ( ! [Z2: 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,Z2),Xd))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xd),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9825_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z2),topolo2177554685111907308n_real(Z2,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),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F4,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9826_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_mh(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_mh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N7))),groups6591440286371151544t_real(aTP_Lamm_mh(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,numeral_numeral_nat(bit0(one))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9827_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_mh(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_mh(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,numeral_numeral_nat(bit0(one))),N7)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_mh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9828_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_bb,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9829_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_9830_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_mi,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9831_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mj(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9832_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X7,N)) )
     => filterlim_nat_real(aTP_Lamm_mk(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9833_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_ml,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9834_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_mm,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9835_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_mn,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9836_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mo,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9837_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N7)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9838_LIMSEQ__realpow__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xd),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xd),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9839_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xd: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mp(Xd),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9840_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_9841_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_9842_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xd)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mq,Xd),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9843_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mr,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9844_tendsto__exp__limit__sequentially,axiom,
    ! [Xd: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ms,Xd),topolo2815343760600316023s_real(aa_real_real(exp_real,Xd)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9845_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mt,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9846_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_mh(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9847_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_mh(A)) ) ) ) ).

% summable
tff(fact_9848_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mu(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mv(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9849_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_mw(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mv(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9850_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_mx(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mh(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9851_zeroseq__arctan__series,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xd)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ci,Xd),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9852_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_mh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Nb)))),suminf_real(aTP_Lamm_mh(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9853_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_mx(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mh(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9854_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_mh(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N7)))),L3)
              & filterlim_nat_real(aTP_Lamm_mx(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_mh(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,numeral_numeral_nat(bit0(one))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_my(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9855_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_my(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mh(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9856_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_my(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_mh(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9857_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_mh(A))),groups6591440286371151544t_real(aTP_Lamm_mh(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,numeral_numeral_nat(bit0(one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9858_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C4: real] :
        ! [X5: real] : aa_real_real(F,X5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X5),C4) ) ).

% real_bounded_linear
tff(fact_9859_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9860_tendsto__exp__limit__at__right,axiom,
    ! [Xd: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_mz,Xd),topolo2815343760600316023s_real(aa_real_real(exp_real,Xd)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9861_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9862_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_9863_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_9864_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))))) ).

% filterlim_tan_at_right
tff(fact_9865_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9866_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_9867_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => aa_set_real_o(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9868_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_9869_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_9870_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9871_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_na(Nb),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9872_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_na(Nb),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9873_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9874_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_9875_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_9876_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_9877_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_9878_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_nb,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9879_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_9880_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_9881_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_nc,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9882_tendsto__exp__limit__at__top,axiom,
    ! [Xd: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_nd,Xd),topolo2815343760600316023s_real(aa_real_real(exp_real,Xd)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9883_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))))) ).

% filterlim_tan_at_left
tff(fact_9884_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_9885_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9886_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xd: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
     => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9887_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_lm(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9888_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_nh(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9889_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_nh(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9890_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_ni(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9891_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_nj(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9892_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_nj(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9893_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_nk(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9894_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_nk(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9895_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_9896_lhopital,axiom,
    ! [F: fun_real_real,Xd: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F2: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,top_top_set_real))
       => ( eventually_real(aTP_Lamm_ne(G),topolo2177554685111907308n_real(Xd,top_top_set_real))
         => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),F2,topolo2177554685111907308n_real(Xd,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),F2,topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9897_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_9898_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_9899_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_9900_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xd: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xd,top_top_set_real))
     => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9901_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Xd: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_ne(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),topolo2815343760600316023s_real(Xd),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G),F),topolo2815343760600316023s_real(Xd),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9902_lhopital__right,axiom,
    ! [F: fun_real_real,Xd: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F2: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
       => ( eventually_real(aTP_Lamm_ne(G),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
         => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),F2,topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),F2,topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9903_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F2: 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_ne(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),F2,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F0),G0),F2,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9904_lhopital__left,axiom,
    ! [F: fun_real_real,Xd: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F2: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
       => ( eventually_real(aTP_Lamm_ne(G),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
         => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),F2,topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),F2,topolo2177554685111907308n_real(Xd,set_or5984915006950818249n_real(Xd))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9905_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_9906_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: 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_nf(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_9907_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xd: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
     => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xd,set_or5849166863359141190n_real(Xd))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9908_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Xd: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_ne(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_nf(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G2),F4),topolo2815343760600316023s_real(Xd),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(G),F),topolo2815343760600316023s_real(Xd),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9909_Bseq__realpow,axiom,
    ! [Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xd)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xd),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9910_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_9911_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),M) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9912_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(inverse_inverse_rat(P3)) = produc4245557441103728435nt_int(aTP_Lamm_nm,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9913_rat__one__code,axiom,
    quotient_of(one_one_rat) = product_Pair_int_int(one_one_int,one_one_int) ).

% rat_one_code
tff(fact_9914_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int) ).

% quotient_of_number(3)
tff(fact_9915_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9916_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int),one_one_int) ).

% quotient_of_number(4)
tff(fact_9917_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(5)
tff(fact_9918_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9919_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9920_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_9921_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = product_Pair_int_int(A,one_one_int) ).

% quotient_of_int
tff(fact_9922_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F) )
       => ( ! [X: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) )
         => ( ! [X: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G) )
           => ( ! [X: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X,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),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9923_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T5: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T5)
               => ( R2 != plus_plus_rat(S2,T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9924_abs__rat__def,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),uminus_uminus_rat(A),A) ).

% abs_rat_def
tff(fact_9925_sgn__rat__def,axiom,
    ! [A: rat] :
      sgn_sgn_rat(A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9926_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9927_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( normalize(product_Pair_int_int(P3,Q2)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9928_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(product_Pair_int_int(P3,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_9929_less__eq__rat__def,axiom,
    ! [Xd: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xd),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xd),Y)
        | ( Xd = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9930_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q2: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9931_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P3: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P3,Q2)) = normalize(product_Pair_int_int(R2,S)) )
         => ( aa_int_int(times_times_int(P3),S) = aa_int_int(times_times_int(R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9932_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(A,zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9933_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(zero_zero_int,A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9934_Frct__code__post_I3_J,axiom,
    frct(product_Pair_int_int(one_one_int,one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9935_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))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9936_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9937_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ? [L3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_9938_set__bits__int__def,axiom,
    ! [F: fun_nat_o] :
      bit_bi6516823479961619367ts_int(F) = $ite(
        ? [N6: nat] :
        ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M4)
         => ( aa_nat_o(F,M4) = aa_nat_o(F,N6) ) ),
        $let(
          n: nat,
          n:= ord_Least_nat(aTP_Lamm_nn(F)),
          aa_int_int(bit_ri631733984087533419it_int(n),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,bit0(one)),map_nat_o(F,upt(zero_zero_nat,aa_nat_nat(suc,n))))) ),
        zero_zero_int ) ).

% set_bits_int_def
tff(fact_9939_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_9940_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_lm(P))) ) ) ) ).

% Least_Suc
tff(fact_9941_Least__Suc2,axiom,
    ! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,M: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(Q,M)
       => ( ~ 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_9942_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_no(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9943_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_9944_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_9945_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9946_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),A2)
           => aa_set_real_o(member_real(aa_real_real(F,X)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_np(F)) ) ) ).

% continuous_on_artanh'
tff(fact_9947_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z2),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9948_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ~ ! [Xi2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi2),B)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F4,Xi2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9949_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9950_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9951_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9952_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,Xd: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xd)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xd),B)
             => ( aa_real_real(F,Xd) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9953_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9954_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                          ( ! [X5: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                                & ! [X5: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9955_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                        ( ! [X5: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                              & ! [X5: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9956_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg,TreeList),Summary)),Deg4)
    <=> ( ( Deg = Deg4 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),numeral_numeral_nat(bit0(one))),
            $let(
              m: nat,
              m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
              ( ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList))
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
              & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
              & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X8)
                    & ! [X5: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg),TreeList),Summary),n),m))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9957_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xd = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                      ( ! [X5: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                            & ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                               => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9958_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                            ( ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                                  & ! [X5: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9959_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                            ( ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                                  & ! [X5: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9960_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xd: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xd),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xd,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xd = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),numeral_numeral_nat(bit0(one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                          ( ! [X5: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X8)
                                & ! [X5: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9961_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_nr(M),bit_take_bit_num(numeral_numeral_nat(M),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9962_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9963_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_9964_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(numeral_numeral_nat(R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9965_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),bit0(M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ns,bit_take_bit_num(Nb,M)) ).

% take_bit_num_simps(3)
tff(fact_9966_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,M))) ).

% take_bit_num_simps(4)
tff(fact_9967_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(numeral_numeral_nat(R2),bit0(M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ns,bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_9968_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(numeral_numeral_nat(R2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_9969_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Nb)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(Nb))) ).

% and_minus_numerals(7)
tff(fact_9970_and__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(Nb))) ).

% and_minus_numerals(3)
tff(fact_9971_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bit0(Nb))) ).

% and_minus_numerals(8)
tff(fact_9972_and__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bit0(Nb))) ).

% and_minus_numerals(4)
tff(fact_9973_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9974_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one) = some_num(bit0(M)) ).

% and_not_num.simps(4)
tff(fact_9975_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,bit0(Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9976_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_9977_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(bit0(M)) ).

% and_not_num.simps(7)
tff(fact_9978_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(M,Nb) = some_num(Q2) )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9979_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,M),bit0(Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_nt,bit_and_not_num(M,Nb)) ).

% and_not_num.simps(8)
tff(fact_9980_and__not__num__eq__None__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( bit_and_not_num(M,Nb) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9981_int__numeral__and__not__num,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,Nb)) ).

% int_numeral_and_not_num
tff(fact_9982_int__numeral__not__and__num,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,M)) ).

% int_numeral_not_and_num
tff(fact_9983_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_nu(M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9984_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,bit0(M)) = case_nat_option_num(none_num,aTP_Lamm_nv(M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9985_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_nw,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9986_take__bit__num__def,axiom,
    ! [Nb: nat,M: num] :
      bit_take_bit_num(Nb,M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),numeral_numeral_nat(M)) = 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),numeral_numeral_nat(M))))) ).

% take_bit_num_def
tff(fact_9987_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,numeral_numeral_nat(Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9988_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_lk),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9989_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_lj),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9990_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9991_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9992_max__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_nx,Nb)),M) ).

% max_Suc1
tff(fact_9993_max__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_ny,Nb)),M) ).

% max_Suc2
tff(fact_9994_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( numeral_numeral_nat(aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9995_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_9996_diff__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ap),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ).

% diff_Suc
tff(fact_9997_min__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_nz,Nb)),M) ).

% min_Suc1
tff(fact_9998_min__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_oa,Nb)),M) ).

% min_Suc2
tff(fact_9999_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)) = bit0(aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_10000_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( 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,M),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_10001_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),inc(aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_10002_set__bits__int__unfold_H,axiom,
    ! [F: fun_nat_o] :
      bit_bi6516823479961619367ts_int(F) = $ite(
        ? [N6: nat] :
        ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
         => ~ aa_nat_o(F,N9) ),
        groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,bit0(one)),map_nat_o(F,upt(zero_zero_nat,ord_Least_nat(aTP_Lamm_ob(F))))),
        $ite(
          ? [N6: nat] :
          ! [N9: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N9)
           => aa_nat_o(F,N9) ),
          $let(
            n: nat,
            n:= ord_Least_nat(aTP_Lamm_oc(F)),
            aa_int_int(bit_ri631733984087533419it_int(n),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,bit0(one)),append_o(map_nat_o(F,upt(zero_zero_nat,n)),cons_o($true,nil_o)))) ),
          zero_zero_int ) ) ).

% set_bits_int_unfold'
tff(fact_10003_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_10004_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_10005_pure__false,axiom,
    pure_assn($false) = bot_bot_assn ).

% pure_false
tff(fact_10006_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot_assn )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_10007_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_10008_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_10009_ent__false__iff,axiom,
    ! [P: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,P),bot_bot_assn)
    <=> ! [H2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H2) ) ).

% ent_false_iff
tff(fact_10010_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_10011_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_10012_ent__false,axiom,
    ! [P: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,bot_bot_assn),P) ).

% ent_false
tff(fact_10013_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(bot_bot_assn),H) ).

% mod_false
tff(fact_10014_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option4927543243414619207at_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((Vd),(Ve))),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V,Va2,Vb),Vc2)) = bot_bot_assn ).

% vebt_assn_raw.simps(4)
tff(fact_10015_vebt__assn__raw_Oelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xd),Xaa) = Y )
     => ( ! [A4: $o,B2: $o] :
            ( ( Xd = vEBT_Leaf((A4),(B2)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A4) )
                      & ( (Bi) = (B2) ) )) ) ) )
       => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2) )
             => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2) )
                 => ( Y != aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(pure_assn(( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_lb(Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [V3: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V3,Va,Vb3),Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xd = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V3: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V3,Va,Vb3),Vc3)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_10016_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ord_less_eq_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72)),char2((C0),(C1),(C22),(C32),(C42),(C52),(C6),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_od,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_od,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_eq_char_simp
tff(fact_10017_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leafi((X21),(X222)) = vEBT_Leafi((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBTi.inject(2)
tff(fact_10018_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_10019_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_10020_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_10021_upt__rec__numeral,axiom,
    ! [M: num,Nb: num] :
      upt(numeral_numeral_nat(M),numeral_numeral_nat(Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Nb)),cons_nat(numeral_numeral_nat(M),upt(aa_nat_nat(suc,numeral_numeral_nat(M)),numeral_numeral_nat(Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_10022_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ord_less_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72)),char2((C0),(C1),(C22),(C32),(C42),(C52),(C6),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_od,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_od,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_char_simp
tff(fact_10023_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10024_list__encode_Ocases,axiom,
    ! [Xd: list_nat] :
      ( ( Xd != nil_nat )
     => ~ ! [X: nat,Xs2: list_nat] : Xd != cons_nat(X,Xs2) ) ).

% list_encode.cases
tff(fact_10025_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,Vd: $o,Ve: $o] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va2,Vb),Vc2)),vEBT_Leafi((Vd),(Ve))) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_10026_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_10027_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_10028_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_10029_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_10030_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X222: $o] : aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14) != vEBT_Leafi((X21),(X222)) ).

% VEBTi.distinct(1)
tff(fact_10031_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] : Y != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X112,X122,X132),X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leafi((X212),(X223)) ) ).

% VEBTi.exhaust
tff(fact_10032_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V739175172307565963ildupi(zero_zero_nat) = heap_T3630416162098727440_VEBTi(vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_10033_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero_nat ).

% VEBTi.size(4)
tff(fact_10034_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero_nat ).

% VEBTi.size_gen(2)
tff(fact_10035_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_10036_vebt__assn__raw_Ocases,axiom,
    ! [Xd: produc3625547720036274456_VEBTi] :
      ( ! [A4: $o,B2: $o,Ai: $o,Bi: $o] : Xd != produc6084888613844515218_VEBTi(vEBT_Leaf((A4),(B2)),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] : Xd != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2))
       => ( ! [V3: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xd != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V3,Va,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V3: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xd != produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V3,Va,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_10037_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Ai2: $o,Bi2: $o] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((A),(B))),vEBT_Leafi((Ai2),(Bi2))) = pure_assn(( ( (Ai2) = (A) )
        & ( (Bi2) = (B) ) )) ).

% vebt_assn_raw.simps(1)
tff(fact_10038_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_10039_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_10040_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(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_10041_less__char__def,axiom,
    ! [C1: char,C22: char] :
      ( ord_less_char(C1,C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_char_def
tff(fact_10042_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)),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% nat_of_char_less_256
tff(fact_10043_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),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% range_nat_of_char
tff(fact_10044_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : size_size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero_nat ).

% char.size(2)
tff(fact_10045_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : size_char(char2((X15),(X2),(X32),(X42),(X52),(X62),(X72),(X82))) = zero_zero_nat ).

% char.size_gen
tff(fact_10046_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10047_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% UNIV_char_of_nat
tff(fact_10048_vebt__assn__raw_Opelims,axiom,
    ! [Xd: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xd),Xaa) = Y )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(Xd,Xaa))
       => ( ! [A4: $o,B2: $o] :
              ( ( Xd = vEBT_Leaf((A4),(B2)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A4) )
                          & ( (Bi) = (B2) ) )) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((A4),(B2)),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2) )
               => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2) )
                   => ( ( Y = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(pure_assn(( ( Mmoi2 = Mmo2 )
                                    & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_lb(Tree_list2),Tree_array2))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2))) ) ) )
           => ( ! [V3: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( Xd = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V3,Va,Vb3),Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V3,Va,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xd = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V3: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V3,Va,Vb3),Vc3) )
                       => ( ( Y = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V3,Va,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_10049_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,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_10050_upto_Opelims,axiom,
    ! [Xd: int,Xaa: int,Y: list_int] :
      ( ( upto(Xd,Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xd,Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),Xaa),cons_int(Xd,upto(aa_int_int(plus_plus_int(Xd),one_one_int),Xaa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xd,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_10051_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_10052_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_10053_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_10054_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_10055_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_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(plus_plus_int(I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_10056_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(J),I)),one_one_int)) ).

% length_upto
tff(fact_10057_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,M),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,M)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_10058_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,M),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,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_10059_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),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,M))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_10060_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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,M))),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,M)),upto(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_10061_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_10062_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_10063_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_10064_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(plus_plus_int(J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_10065_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(minus_minus_int(J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_10066_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(minus_minus_int(J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_10067_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_10068_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(plus_plus_int(I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_10069_upto_Oelims,axiom,
    ! [Xd: int,Xaa: int,Y: list_int] :
      ( ( upto(Xd,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xd),Xaa),cons_int(Xd,upto(aa_int_int(plus_plus_int(Xd),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_10070_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(plus_plus_int(I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_10071_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(minus_minus_int(J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_10072_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(plus_plus_int(I),one_one_int),aa_int_int(minus_minus_int(J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_10073_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(minus_minus_int(J),one_one_int)),cons_int(J,upto(aa_int_int(plus_plus_int(J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_10074_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(plus_plus_int(I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_10075_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_10076_sorted__list__of__set__range,axiom,
    ! [M: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) = upt(M,Nb) ).

% sorted_list_of_set_range
tff(fact_10077_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_10078_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_10079_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_10080_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_10081_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_10082_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U2)
     => ( ! [I3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),U2) )
           => ~ aa_nat_o(P,I3) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U2)) ) ) ) ).

% upt_filter_extend
tff(fact_10083_sort__upt,axiom,
    ! [M: nat,Nb: nat] : linord738340561235409698at_nat(aTP_Lamm_ap,upt(M,Nb)) = upt(M,Nb) ).

% sort_upt
tff(fact_10084_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_aq,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_10085_tl__upt,axiom,
    ! [M: nat,Nb: nat] : tl_nat(upt(M,Nb)) = upt(aa_nat_nat(suc,M),Nb) ).

% tl_upt
tff(fact_10086_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_10087_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one)))) ).

% csqrt.simps(1)
tff(fact_10088_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).

% complex_Re_numeral
tff(fact_10089_Re__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Re_divide_of_nat
tff(fact_10090_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R2) ).

% Re_divide_of_real
tff(fact_10091_Re__sgn,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Re_sgn
tff(fact_10092_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_10093_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( aa_complex_real(re,Y) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Y)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_10094_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xd: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),Xd)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xd)) ).

% scaleR_complex.simps(1)
tff(fact_10095_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_10096_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_10097_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_10098_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_10099_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_10100_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_10101_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one)))))) ).

% csqrt.code
tff(fact_10102_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(bit0(one))))) ).

% csqrt.simps(2)
tff(fact_10103_complex__Im__fact,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,Nb)) = zero_zero_real ).

% complex_Im_fact
tff(fact_10104_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_10105_Im__complex__of__real,axiom,
    ! [Z: real] : aa_complex_real(im,aa_real_complex(real_V4546457046886955230omplex,Z)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_10106_Im__power__real,axiom,
    ! [Xd: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xd) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(Xd),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_10107_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_10108_complex__Im__of__nat,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,Nb)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_10109_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R2) ).

% Im_divide_of_real
tff(fact_10110_Im__sgn,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Im_sgn
tff(fact_10111_Re__power__real,axiom,
    ! [Xd: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xd) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(Xd),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xd)),Nb) ) ) ).

% Re_power_real
tff(fact_10112_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_10113_Im__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Im_divide_of_nat
tff(fact_10114_csqrt__of__real__nonneg,axiom,
    ! [Xd: complex] :
      ( ( aa_complex_real(im,Xd) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xd))
       => ( aa_complex_complex(csqrt,Xd) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xd))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_10115_csqrt__minus,axiom,
    ! [Xd: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xd)),zero_zero_real)
        | ( ( aa_complex_real(im,Xd) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xd)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xd)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xd)) ) ) ).

% csqrt_minus
tff(fact_10116_csqrt__of__real__nonpos,axiom,
    ! [Xd: complex] :
      ( ( aa_complex_real(im,Xd) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xd)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xd) = 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,Xd))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_10117_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I2: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I2) ) ) ).

% complex_is_Int_iff
tff(fact_10118_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_10119_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_10120_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_10121_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xd: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),Xd)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xd)) ).

% scaleR_complex.simps(2)
tff(fact_10122_times__complex_Osimps_I2_J,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xd)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_10123_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(im,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(re,Z)) ) ) ).

% cmod_eq_Re
tff(fact_10124_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(re,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(im,Z)) ) ) ).

% cmod_eq_Im
tff(fact_10125_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z)) = aa_complex_real(real_V1022390504157884413omplex,Z) )
     => ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_10126_times__complex_Osimps_I1_J,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xd),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_10127_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xd: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),Xd) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xd))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xd))) ).

% scaleR_complex.code
tff(fact_10128_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_10129_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_10130_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_10131_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_10132_times__complex_Ocode,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(times_times_complex(Xd),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,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),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,Xd)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_10133_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),numeral_numeral_nat(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),numeral_numeral_nat(bit0(one)))) ).

% cmod_power2
tff(fact_10134_Im__power2,axiom,
    ! [Xd: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(bit0(one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,Xd))),aa_complex_real(im,Xd)) ).

% Im_power2
tff(fact_10135_Re__power2,axiom,
    ! [Xd: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xd),numeral_numeral_nat(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,Xd)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xd)),numeral_numeral_nat(bit0(one)))) ).

% Re_power2
tff(fact_10136_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),numeral_numeral_nat(bit0(one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_10137_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),numeral_numeral_nat(bit0(one))))) ).

% norm_complex_def
tff(fact_10138_inverse__complex_Osimps_I1_J,axiom,
    ! [Xd: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xd)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xd)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xd)),numeral_numeral_nat(bit0(one))))) ).

% inverse_complex.simps(1)
tff(fact_10139_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),numeral_numeral_nat(bit0(one))))) ) ).

% complex_neq_0
tff(fact_10140_Re__divide,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xd),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,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),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)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),numeral_numeral_nat(bit0(one))))) ).

% Re_divide
tff(fact_10141_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),numeral_numeral_nat(bit0(one)))) = B ) ) ).

% csqrt_square
tff(fact_10142_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),numeral_numeral_nat(bit0(one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_10143_inverse__complex_Osimps_I2_J,axiom,
    ! [Xd: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xd)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xd)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xd)),numeral_numeral_nat(bit0(one))))) ).

% inverse_complex.simps(2)
tff(fact_10144_Im__divide,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xd),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,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xd)),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)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),numeral_numeral_nat(bit0(one))))) ).

% Im_divide
tff(fact_10145_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_10146_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),numeral_numeral_nat(bit0(one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_10147_inverse__complex_Ocode,axiom,
    ! [Xd: complex] : aa_complex_complex(invers8013647133539491842omplex,Xd) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xd)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xd)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xd)),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xd))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xd)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xd)),numeral_numeral_nat(bit0(one)))))) ).

% inverse_complex.code
tff(fact_10148_Complex__divide,axiom,
    ! [Xd: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xd),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,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xd)),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)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),numeral_numeral_nat(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,Xd)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xd)),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)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),numeral_numeral_nat(bit0(one)))))) ).

% Complex_divide
tff(fact_10149_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),numeral_numeral_nat(bit0(one)))) ) ) ).

% Im_Reals_divide
tff(fact_10150_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),numeral_numeral_nat(bit0(one)))) ) ) ).

% Re_Reals_divide
tff(fact_10151_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R2)) ) ) ).

% Re_divide_Reals
tff(fact_10152_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xd: complex] :
      ( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xd),real_V2521375963428798218omplex)
       => ( ( Xd = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( Xd = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_10153_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xd: complex] :
      ( aa_set_complex_o(member_complex(Y),real_V2521375963428798218omplex)
     => ( aa_set_complex_o(member_complex(Xd),real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = Xd )
        <=> ( ( Xd = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_10154_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R2),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R2)) ) ) ).

% Im_divide_Reals
tff(fact_10155_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),real_V2521375963428798218omplex)
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_10156_Complex__in__Reals,axiom,
    ! [Xd: real] : aa_set_complex_o(member_complex(aa_real_complex(complex2(Xd),zero_zero_real)),real_V2521375963428798218omplex) ).

% Complex_in_Reals
tff(fact_10157_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),numeral_numeral_nat(bit0(one))))) ).

% complex_mult_cnj
tff(fact_10158_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_10159_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_10160_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_10161_complex__cnj__zero,axiom,
    aa_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

% complex_cnj_zero
tff(fact_10162_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
tff(fact_10163_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_10164_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_10165_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_oe,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_10166_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = numeral_numeral_nat(bit0(one)) ).

% card_UNIV_bool
tff(fact_10167_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_10168_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_10169_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_10170_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_10171_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),aa_int_int(plus_plus_int(L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_10172_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_10173_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_of(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_og(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_10174_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_of(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_og(M7),I))) ) ) ).

% card_less_Suc
tff(fact_10175_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_og(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_10176_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_10177_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_10178_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_10179_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_10180_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_10181_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))) ).

% card_UNIV_char
tff(fact_10182_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),S3)) ).

% card_sum_le_nat_sum
tff(fact_10183_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_kz(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_10184_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_bh(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_10185_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_10186_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_10187_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_10188_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_10189_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_10190_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_10191_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_10192_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_10193_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),numeral_numeral_nat(bit0(one))) ).

% complex_mod_mult_cnj
tff(fact_10194_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_10195_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),numeral_numeral_nat(bit0(one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_10196_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_10197_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_10198_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),numeral_numeral_nat(bit0(one))))) ).

% complex_div_cnj
tff(fact_10199_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_10200_card__num0,axiom,
    finite6454714172617411596l_num0(top_to3689904424835650196l_num0) = zero_zero_nat ).

% card_num0
tff(fact_10201_card__num1,axiom,
    finite6454714172617411597l_num1(top_to3689904429138878997l_num1) = one_one_nat ).

% card_num1
tff(fact_10202_card__nat,axiom,
    aa_set_nat_nat(finite_card_nat,top_top_set_nat) = zero_zero_nat ).

% card_nat
tff(fact_10203_card__literal,axiom,
    finite_card_literal(top_top_set_literal) = zero_zero_nat ).

% card_literal
tff(fact_10204_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_10205_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_10206_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jm,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),numeral_numeral_nat(bit0(one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10207_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jq,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10208_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_kx,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),numeral_numeral_real(bit0(one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10209_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jw,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10210_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mg,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10211_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_nb,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10212_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_mi,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10213_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ld,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_10214_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ml,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_9
tff(fact_10215_ATP_Olambda__10,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_la,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Uu) ) ).

% ATP.lambda_10
tff(fact_10216_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_lz,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ji,Uu)) ).

% ATP.lambda_11
tff(fact_10217_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_mm,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_12
tff(fact_10218_ATP_Olambda__13,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_nt,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_13
tff(fact_10219_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_ns,Uu) = some_num(bit0(Uu)) ).

% ATP.lambda_14
tff(fact_10220_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ks,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_15
tff(fact_10221_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_nw,Uu) = some_num(one) ).

% ATP.lambda_16
tff(fact_10222_ATP_Olambda__17,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_jr(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_17
tff(fact_10223_ATP_Olambda__18,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_nm,Uu),Uua) = $ite(Uu = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,Uu)),Uua),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_18
tff(fact_10224_ATP_Olambda__19,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_nv(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_ns,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_19
tff(fact_10225_ATP_Olambda__20,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_20
tff(fact_10226_ATP_Olambda__21,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_21
tff(fact_10227_ATP_Olambda__22,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))) ).

% ATP.lambda_22
tff(fact_10228_ATP_Olambda__23,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ky,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_23
tff(fact_10229_ATP_Olambda__24,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ku,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(bit0(one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_24
tff(fact_10230_ATP_Olambda__25,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_kt(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_25
tff(fact_10231_ATP_Olambda__26,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ji,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))) ).

% ATP.lambda_26
tff(fact_10232_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ma,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_27
tff(fact_10233_ATP_Olambda__28,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_28
tff(fact_10234_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_km,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),numeral_numeral_nat(bit0(one))) ).

% ATP.lambda_29
tff(fact_10235_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_30
tff(fact_10236_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_nc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_31
tff(fact_10237_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_32
tff(fact_10238_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ke,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_33
tff(fact_10239_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_bh(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_34
tff(fact_10240_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_bg,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_35
tff(fact_10241_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ms,Uu),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,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_36
tff(fact_10242_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_mz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_37
tff(fact_10243_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_nd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_38
tff(fact_10244_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ci,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat))) ).

% ATP.lambda_39
tff(fact_10245_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kw,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_40
tff(fact_10246_ATP_Olambda__41,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_el(Uu),Uua) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(aa_nat_Code_integer(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_Code_integer(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_41
tff(fact_10247_ATP_Olambda__42,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_em(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_complex(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_42
tff(fact_10248_ATP_Olambda__43,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_ej(Uu),Uua) = aa_assn_assn(times_times_assn(aa_nat_assn(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_assn(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_43
tff(fact_10249_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ek(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_44
tff(fact_10250_ATP_Olambda__45,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_en(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_45
tff(fact_10251_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_eo(Uu),Uua) = aa_int_int(times_times_int(aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))) ).

% ATP.lambda_46
tff(fact_10252_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_iz(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_47
tff(fact_10253_ATP_Olambda__48,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iy(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_48
tff(fact_10254_ATP_Olambda__49,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_id(Uu),Uua)
    <=> ( aa_Code_integer_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_49
tff(fact_10255_ATP_Olambda__50,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_ig(Uu),Uua)
    <=> ( aa_Code_integer_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_50
tff(fact_10256_ATP_Olambda__51,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_ij(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_51
tff(fact_10257_ATP_Olambda__52,axiom,
    ! [Uu: fun_complex_assn,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ic(Uu),Uua)
    <=> ( aa_complex_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_52
tff(fact_10258_ATP_Olambda__53,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_if(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_53
tff(fact_10259_ATP_Olambda__54,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ii(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_54
tff(fact_10260_ATP_Olambda__55,axiom,
    ! [Uu: fun_int_assn,Uua: int] :
      ( aa_int_o(aTP_Lamm_ib(Uu),Uua)
    <=> ( aa_int_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_55
tff(fact_10261_ATP_Olambda__56,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ie(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_56
tff(fact_10262_ATP_Olambda__57,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ih(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_57
tff(fact_10263_ATP_Olambda__58,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ik(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_58
tff(fact_10264_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_my(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_mh(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_59
tff(fact_10265_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mx(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_mh(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))) ).

% ATP.lambda_60
tff(fact_10266_ATP_Olambda__61,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_nr(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(Uu)),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),numeral_numeral_nat(Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_61
tff(fact_10267_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ll,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_62
tff(fact_10268_ATP_Olambda__63,axiom,
    ! [Uu: $o,Uua: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_od,(Uu)),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))) ).

% ATP.lambda_63
tff(fact_10269_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_64
tff(fact_10270_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_65
tff(fact_10271_ATP_Olambda__66,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_de(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(bit0(one)))) ).

% ATP.lambda_66
tff(fact_10272_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dd,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(bit0(one)))) ).

% ATP.lambda_67
tff(fact_10273_ATP_Olambda__68,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dc(Uu),Uua) = plus_plus_rat(Uu,divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(bit0(one)))) ).

% ATP.lambda_68
tff(fact_10274_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_69
tff(fact_10275_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_70
tff(fact_10276_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_71
tff(fact_10277_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_72
tff(fact_10278_ATP_Olambda__73,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_eb(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_73
tff(fact_10279_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ea,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_74
tff(fact_10280_ATP_Olambda__75,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dz(Uu),Uua) = plus_plus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_75
tff(fact_10281_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ec,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_76
tff(fact_10282_ATP_Olambda__77,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ed(Uu),Uua) = aa_int_int(plus_plus_int(Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_77
tff(fact_10283_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_oe,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_78
tff(fact_10284_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_li,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_79
tff(fact_10285_ATP_Olambda__80,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cf,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_80
tff(fact_10286_ATP_Olambda__81,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_bd(Uu),Uua) = aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(Uua),Uu) ).

% ATP.lambda_81
tff(fact_10287_ATP_Olambda__82,axiom,
    ! [Uu: complex,Uua: complex] : aa_complex_complex(aTP_Lamm_be(Uu),Uua) = aa_complex_complex(times_times_complex(Uua),Uu) ).

% ATP.lambda_82
tff(fact_10288_ATP_Olambda__83,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aTP_Lamm_az(Uu),Uua) = aa_assn_assn(times_times_assn(Uua),Uu) ).

% ATP.lambda_83
tff(fact_10289_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ba,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_84
tff(fact_10290_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_85
tff(fact_10291_ATP_Olambda__86,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_bc(Uu),Uua) = aa_int_int(times_times_int(Uua),Uu) ).

% ATP.lambda_86
tff(fact_10292_ATP_Olambda__87,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_87
tff(fact_10293_ATP_Olambda__88,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ln,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_88
tff(fact_10294_ATP_Olambda__89,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_lf(Uu),Uua) = plus_p5714425477246183910nteger(Uua,Uu) ).

% ATP.lambda_89
tff(fact_10295_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_90
tff(fact_10296_ATP_Olambda__91,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_le(Uu),Uua) = aa_int_int(plus_plus_int(Uua),Uu) ).

% ATP.lambda_91
tff(fact_10297_ATP_Olambda__92,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_92
tff(fact_10298_ATP_Olambda__93,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_aj(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_93
tff(fact_10299_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ak,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_94
tff(fact_10300_ATP_Olambda__95,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_al,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_95
tff(fact_10301_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kd,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_96
tff(fact_10302_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mn,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_97
tff(fact_10303_ATP_Olambda__98,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_98
tff(fact_10304_ATP_Olambda__99,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_99
tff(fact_10305_ATP_Olambda__100,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_100
tff(fact_10306_ATP_Olambda__101,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_101
tff(fact_10307_ATP_Olambda__102,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_102
tff(fact_10308_ATP_Olambda__103,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_103
tff(fact_10309_ATP_Olambda__104,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_cv(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_104
tff(fact_10310_ATP_Olambda__105,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ct(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua)) ) ).

% ATP.lambda_105
tff(fact_10311_ATP_Olambda__106,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cn(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_106
tff(fact_10312_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_cp(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_107
tff(fact_10313_ATP_Olambda__108,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_cr(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_108
tff(fact_10314_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ka(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_109
tff(fact_10315_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jz(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)) ).

% ATP.lambda_110
tff(fact_10316_ATP_Olambda__111,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_nk(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_111
tff(fact_10317_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_dq(Uu),Uua) = aa_nat_Code_integer(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_112
tff(fact_10318_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dr(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_113
tff(fact_10319_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_do(Uu),Uua) = aa_nat_assn(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_114
tff(fact_10320_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dp(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_115
tff(fact_10321_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_lm(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_116
tff(fact_10322_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_di(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_117
tff(fact_10323_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_dj(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_118
tff(fact_10324_ATP_Olambda__119,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_nu(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_119
tff(fact_10325_ATP_Olambda__120,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_md(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_mc(Uu),Uua)) ).

% ATP.lambda_120
tff(fact_10326_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ko(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_121
tff(fact_10327_ATP_Olambda__122,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ne(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_122
tff(fact_10328_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bw,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Uua))) ) ).

% ATP.lambda_123
tff(fact_10329_ATP_Olambda__124,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_by(Uu),Uua) = times_3573771949741848930nteger(plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_124
tff(fact_10330_ATP_Olambda__125,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_cb(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_125
tff(fact_10331_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bz(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_126
tff(fact_10332_ATP_Olambda__127,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bx(Uu),Uua) = times_times_rat(plus_plus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_127
tff(fact_10333_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cc(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_128
tff(fact_10334_ATP_Olambda__129,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ca(Uu),Uua) = times_times_int(aa_int_int(plus_plus_int(Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_129
tff(fact_10335_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mq,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_130
tff(fact_10336_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_131
tff(fact_10337_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oa,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_132
tff(fact_10338_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nx,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_133
tff(fact_10339_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_134
tff(fact_10340_ATP_Olambda__135,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mk(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_135
tff(fact_10341_ATP_Olambda__136,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_iw(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_136
tff(fact_10342_ATP_Olambda__137,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ix(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_137
tff(fact_10343_ATP_Olambda__138,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_dv(Uu),Uua) = times_3573771949741848930nteger(aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_138
tff(fact_10344_ATP_Olambda__139,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_dw(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_139
tff(fact_10345_ATP_Olambda__140,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_fun_assn_assn(aTP_Lamm_dt(Uu),Uua) = times_times_assn(aa_nat_assn(Uu,Uua)) ).

% ATP.lambda_140
tff(fact_10346_ATP_Olambda__141,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_du(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_141
tff(fact_10347_ATP_Olambda__142,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ds(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_142
tff(fact_10348_ATP_Olambda__143,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_dx(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_143
tff(fact_10349_ATP_Olambda__144,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_dy(Uu),Uua) = times_times_int(aa_nat_int(Uu,Uua)) ).

% ATP.lambda_144
tff(fact_10350_ATP_Olambda__145,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_np(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_145
tff(fact_10351_ATP_Olambda__146,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_no(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_146
tff(fact_10352_ATP_Olambda__147,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mw(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_147
tff(fact_10353_ATP_Olambda__148,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_mf(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_148
tff(fact_10354_ATP_Olambda__149,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_nn(Uu),Uua)
    <=> ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),M4)
         => ( aa_nat_o(Uu,M4)
          <=> aa_nat_o(Uu,Uua) ) ) ) ).

% ATP.lambda_149
tff(fact_10355_ATP_Olambda__150,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_oc(Uu),Uua)
    <=> ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N9)
         => aa_nat_o(Uu,N9) ) ) ).

% ATP.lambda_150
tff(fact_10356_ATP_Olambda__151,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ob(Uu),Uua)
    <=> ! [N9: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N9)
         => ~ aa_nat_o(Uu,N9) ) ) ).

% ATP.lambda_151
tff(fact_10357_ATP_Olambda__152,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_js(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),numeral_numeral_nat(bit0(one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_152
tff(fact_10358_ATP_Olambda__153,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jy(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_153
tff(fact_10359_ATP_Olambda__154,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kr(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(bit0(one))),Uua),Uub)) ).

% ATP.lambda_154
tff(fact_10360_ATP_Olambda__155,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_kp(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),numeral_numeral_nat(Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_155
tff(fact_10361_ATP_Olambda__156,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kq(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Uua)),one_one_int),aa_int_int(minus_minus_int(Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one))),Uua),Uub)) ).

% ATP.lambda_156
tff(fact_10362_ATP_Olambda__157,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_ff(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_157
tff(fact_10363_ATP_Olambda__158,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_fl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_real(Uua,Uub),one_one_real) ).

% ATP.lambda_158
tff(fact_10364_ATP_Olambda__159,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_fj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_159
tff(fact_10365_ATP_Olambda__160,axiom,
    ! [Uu: complex,Uua: fun_complex_assn,Uub: complex] :
      aa_complex_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_fi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_160
tff(fact_10366_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: fun_real_assn,Uub: real] :
      aa_real_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_fe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_161
tff(fact_10367_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_162
tff(fact_10368_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_fg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_163
tff(fact_10369_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_164
tff(fact_10370_ATP_Olambda__165,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_fh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_165
tff(fact_10371_ATP_Olambda__166,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_166
tff(fact_10372_ATP_Olambda__167,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_assn(aa_fun7127563710338807932T_assn(aTP_Lamm_ev(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_167
tff(fact_10373_ATP_Olambda__168,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_fb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_real(Uua,Uub),one_one_real) ).

% ATP.lambda_168
tff(fact_10374_ATP_Olambda__169,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_ez(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_169
tff(fact_10375_ATP_Olambda__170,axiom,
    ! [Uu: complex,Uua: fun_complex_assn,Uub: complex] :
      aa_complex_assn(aa_fun8818621774317227492x_assn(aTP_Lamm_ey(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_170
tff(fact_10376_ATP_Olambda__171,axiom,
    ! [Uu: real,Uua: fun_real_assn,Uub: real] :
      aa_real_assn(aa_fun2649397157800988384l_assn(aTP_Lamm_eu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_171
tff(fact_10377_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fa(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_172
tff(fact_10378_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_ew(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_173
tff(fact_10379_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_174
tff(fact_10380_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_ex(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_175
tff(fact_10381_ATP_Olambda__176,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_176
tff(fact_10382_ATP_Olambda__177,axiom,
    ! [Uu: fun_VEBT_VEBT_assn,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_assn(aa_fun8737571391773076682T_assn(aTP_Lamm_ho(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_177
tff(fact_10383_ATP_Olambda__178,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun3597767889293385422T_real(aTP_Lamm_hy(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_real(Uu,Uub),one_one_real) ).

% ATP.lambda_178
tff(fact_10384_ATP_Olambda__179,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_hw(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_179
tff(fact_10385_ATP_Olambda__180,axiom,
    ! [Uu: fun_complex_assn,Uua: fun_complex_o,Uub: complex] :
      aa_complex_assn(aa_fun6270938053774919130x_assn(aTP_Lamm_hu(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_180
tff(fact_10386_ATP_Olambda__181,axiom,
    ! [Uu: fun_real_assn,Uua: fun_real_o,Uub: real] :
      aa_real_assn(aa_fun3688174147639456858l_assn(aTP_Lamm_hm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_181
tff(fact_10387_ATP_Olambda__182,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_hx(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),one_one_real) ).

% ATP.lambda_182
tff(fact_10388_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_assn,Uua: fun_nat_o,Uub: nat] :
      aa_nat_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_hq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_183
tff(fact_10389_ATP_Olambda__184,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
      aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_hz(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu,Uub),one_one_real) ).

% ATP.lambda_184
tff(fact_10390_ATP_Olambda__185,axiom,
    ! [Uu: fun_int_assn,Uua: fun_int_o,Uub: int] :
      aa_int_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_hs(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_185
tff(fact_10391_ATP_Olambda__186,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_ia(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),one_one_real) ).

% ATP.lambda_186
tff(fact_10392_ATP_Olambda__187,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_nf(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_187
tff(fact_10393_ATP_Olambda__188,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_me(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_188
tff(fact_10394_ATP_Olambda__189,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ls(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_189
tff(fact_10395_ATP_Olambda__190,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_lt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_190
tff(fact_10396_ATP_Olambda__191,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_lis5319508155010942429i_assn(aTP_Lamm_jo(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Uu,Uua)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one)))))),Uua))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one)))))),Uub)) ).

% ATP.lambda_191
tff(fact_10397_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_192
tff(fact_10398_ATP_Olambda__193,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_193
tff(fact_10399_ATP_Olambda__194,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_194
tff(fact_10400_ATP_Olambda__195,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua) ) ) ).

% ATP.lambda_195
tff(fact_10401_ATP_Olambda__196,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_196
tff(fact_10402_ATP_Olambda__197,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Uub)),Uua) ) ) ).

% ATP.lambda_197
tff(fact_10403_ATP_Olambda__198,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_198
tff(fact_10404_ATP_Olambda__199,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_199
tff(fact_10405_ATP_Olambda__200,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_200
tff(fact_10406_ATP_Olambda__201,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_201
tff(fact_10407_ATP_Olambda__202,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_202
tff(fact_10408_ATP_Olambda__203,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_203
tff(fact_10409_ATP_Olambda__204,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_204
tff(fact_10410_ATP_Olambda__205,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_205
tff(fact_10411_ATP_Olambda__206,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_206
tff(fact_10412_ATP_Olambda__207,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_207
tff(fact_10413_ATP_Olambda__208,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_of(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_208
tff(fact_10414_ATP_Olambda__209,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_lb(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(Uua,Uub)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu,Uub)) ).

% ATP.lambda_209
tff(fact_10415_ATP_Olambda__210,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_og(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_210
tff(fact_10416_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_211
tff(fact_10417_ATP_Olambda__212,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_212
tff(fact_10418_ATP_Olambda__213,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_213
tff(fact_10419_ATP_Olambda__214,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_214
tff(fact_10420_ATP_Olambda__215,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_215
tff(fact_10421_ATP_Olambda__216,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_216
tff(fact_10422_ATP_Olambda__217,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hn(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_217
tff(fact_10423_ATP_Olambda__218,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_hv(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_218
tff(fact_10424_ATP_Olambda__219,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ht(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_219
tff(fact_10425_ATP_Olambda__220,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_hl(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_220
tff(fact_10426_ATP_Olambda__221,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_hp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_221
tff(fact_10427_ATP_Olambda__222,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_222
tff(fact_10428_ATP_Olambda__223,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun677759971260770784VEBT_o(aTP_Lamm_fz(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_223
tff(fact_10429_ATP_Olambda__224,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_gl(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_224
tff(fact_10430_ATP_Olambda__225,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun144181944155979022eger_o(aTP_Lamm_gh(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod5621350732091595593omplex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_225
tff(fact_10431_ATP_Olambda__226,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_gf(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_226
tff(fact_10432_ATP_Olambda__227,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_fx(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_227
tff(fact_10433_ATP_Olambda__228,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_gj(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_228
tff(fact_10434_ATP_Olambda__229,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_gb(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_229
tff(fact_10435_ATP_Olambda__230,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_gn(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_230
tff(fact_10436_ATP_Olambda__231,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_gd(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_231
tff(fact_10437_ATP_Olambda__232,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_gp(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_232
tff(fact_10438_ATP_Olambda__233,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5142337294461892954VEBT_o(aTP_Lamm_hf(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_233
tff(fact_10439_ATP_Olambda__234,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_234
tff(fact_10440_ATP_Olambda__235,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aTP_Lamm_hb(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_235
tff(fact_10441_ATP_Olambda__236,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_236
tff(fact_10442_ATP_Olambda__237,axiom,
    ! [Uu: set_real,Uua: fun_real_assn,Uub: real] :
      ( aa_real_o(aa_fun7580339382299851254real_o(aTP_Lamm_hd(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_237
tff(fact_10443_ATP_Olambda__238,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_gr(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_238
tff(fact_10444_ATP_Olambda__239,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aTP_Lamm_hh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_239
tff(fact_10445_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_240
tff(fact_10446_ATP_Olambda__241,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aTP_Lamm_hj(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_241
tff(fact_10447_ATP_Olambda__242,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_242
tff(fact_10448_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_lh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_243
tff(fact_10449_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_244
tff(fact_10450_ATP_Olambda__245,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_kz(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_245
tff(fact_10451_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_mv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(bit0(one))),pi))) ).

% ATP.lambda_246
tff(fact_10452_ATP_Olambda__247,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_247
tff(fact_10453_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_248
tff(fact_10454_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_249
tff(fact_10455_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_250
tff(fact_10456_ATP_Olambda__251,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_251
tff(fact_10457_ATP_Olambda__252,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_252
tff(fact_10458_ATP_Olambda__253,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_253
tff(fact_10459_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_fs(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_254
tff(fact_10460_ATP_Olambda__255,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(Uu),Uua),Uub) = aa_int_int(times_times_int(aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_255
tff(fact_10461_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_256
tff(fact_10462_ATP_Olambda__257,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_lr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_257
tff(fact_10463_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_df(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_258
tff(fact_10464_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_dg(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_259
tff(fact_10465_ATP_Olambda__260,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dh(Uu),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_260
tff(fact_10466_ATP_Olambda__261,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lo(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_261
tff(fact_10467_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_262
tff(fact_10468_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_fv(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_263
tff(fact_10469_ATP_Olambda__264,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_fw(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_264
tff(fact_10470_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_na(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_265
tff(fact_10471_ATP_Olambda__266,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_266
tff(fact_10472_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_267
tff(fact_10473_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ju(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_268
tff(fact_10474_ATP_Olambda__269,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_nj(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_10475_ATP_Olambda__270,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_eg(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_270
tff(fact_10476_ATP_Olambda__271,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ef(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_271
tff(fact_10477_ATP_Olambda__272,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ee(Uu),Uua),Uub) = plus_plus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_272
tff(fact_10478_ATP_Olambda__273,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_eh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_273
tff(fact_10479_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ei(Uu),Uua),Uub) = aa_int_int(plus_plus_int(Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_274
tff(fact_10480_ATP_Olambda__275,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uu,Uub))) ) ).

% ATP.lambda_275
tff(fact_10481_ATP_Olambda__276,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uu,Uub))) ) ).

% ATP.lambda_276
tff(fact_10482_ATP_Olambda__277,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_cx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_277
tff(fact_10483_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_278
tff(fact_10484_ATP_Olambda__279,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_279
tff(fact_10485_ATP_Olambda__280,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_et(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_280
tff(fact_10486_ATP_Olambda__281,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_es(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_281
tff(fact_10487_ATP_Olambda__282,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set_nat_o(Uua,Uub),aa_set_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_282
tff(fact_10488_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_eq(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_283
tff(fact_10489_ATP_Olambda__284,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_er(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_284
tff(fact_10490_ATP_Olambda__285,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> 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,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_285
tff(fact_10491_ATP_Olambda__286,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> 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,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub))) ) ).

% ATP.lambda_286
tff(fact_10492_ATP_Olambda__287,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1332161875654541722_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> 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,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_287
tff(fact_10493_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> 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,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_288
tff(fact_10494_ATP_Olambda__289,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> 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,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_289
tff(fact_10495_ATP_Olambda__290,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ni(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_290
tff(fact_10496_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nh(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_291
tff(fact_10497_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dk(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_292
tff(fact_10498_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_dl(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_293
tff(fact_10499_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mu(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_294
tff(fact_10500_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_ce(Uu),Uua),Uub) = pure_assn(Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua)) ).

% ATP.lambda_295
tff(fact_10501_ATP_Olambda__296,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_cd(Uu),Uua),Uub) = pure_assn(Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua)) ).

% ATP.lambda_296
tff(fact_10502_ATP_Olambda__297,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: fun_VEBT_VEBT_assn,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_assn(aa_fun7127563710338807932T_assn(aa_fun7778866309842214326T_assn(aTP_Lamm_il(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_assn(Uua,Uuc),aa_VEBT_VEBT_assn(Uub,Uuc)) ).

% ATP.lambda_297
tff(fact_10503_ATP_Olambda__298,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ir(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_298
tff(fact_10504_ATP_Olambda__299,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_in(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_assn(Uua,Uuc),aa_Code_integer_assn(Uub,Uuc)) ).

% ATP.lambda_299
tff(fact_10505_ATP_Olambda__300,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_it(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_real(Uua,Uuc),aa_Code_integer_real(Uub,Uuc)) ).

% ATP.lambda_300
tff(fact_10506_ATP_Olambda__301,axiom,
    ! [Uu: complex,Uua: fun_complex_assn,Uub: fun_complex_assn,Uuc: complex] :
      aa_complex_assn(aa_fun8818621774317227492x_assn(aa_fun9098329248302834228x_assn(aTP_Lamm_im(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_assn(Uua,Uuc),aa_complex_assn(Uub,Uuc)) ).

% ATP.lambda_301
tff(fact_10507_ATP_Olambda__302,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_is(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_302
tff(fact_10508_ATP_Olambda__303,axiom,
    ! [Uu: real,Uua: fun_real_assn,Uub: fun_real_assn,Uuc: real] :
      aa_real_assn(aa_fun2649397157800988384l_assn(aa_fun1876152720653945902l_assn(aTP_Lamm_io(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_assn(Uua,Uuc),aa_real_assn(Uub,Uuc)) ).

% ATP.lambda_303
tff(fact_10509_ATP_Olambda__304,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_iu(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ).

% ATP.lambda_304
tff(fact_10510_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_iq(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ).

% ATP.lambda_305
tff(fact_10511_ATP_Olambda__306,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ip(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ).

% ATP.lambda_306
tff(fact_10512_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_kl(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_kk(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_307
tff(fact_10513_ATP_Olambda__308,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ly(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_308
tff(fact_10514_ATP_Olambda__309,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_309
tff(fact_10515_ATP_Olambda__310,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_310
tff(fact_10516_ATP_Olambda__311,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lu(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_311
tff(fact_10517_ATP_Olambda__312,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ki(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_312
tff(fact_10518_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_kh(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_313
tff(fact_10519_ATP_Olambda__314,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: fun_VEBT_VEBT_assn,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5142337294461892954VEBT_o(aa_fun876682331325704374VEBT_o(aTP_Lamm_hg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(Uua,Uuc)),aa_VEBT_VEBT_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_314
tff(fact_10520_ATP_Olambda__315,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_gu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(Uua,Uuc)),aa_VEBT_VEBT_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_315
tff(fact_10521_ATP_Olambda__316,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_hc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_Code_integer_rat(Uua,Uuc)),aa_Code_integer_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_316
tff(fact_10522_ATP_Olambda__317,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_ha(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_317
tff(fact_10523_ATP_Olambda__318,axiom,
    ! [Uu: set_real,Uua: fun_real_assn,Uub: fun_real_assn,Uuc: real] :
      ( aa_real_o(aa_fun7580339382299851254real_o(aa_fun4959323011771448382real_o(aTP_Lamm_he(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_assn_assn(times_times_assn(aa_real_assn(Uua,Uuc)),aa_real_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_318
tff(fact_10524_ATP_Olambda__319,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_gs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_319
tff(fact_10525_ATP_Olambda__320,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_hi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_assn_assn(times_times_assn(aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_320
tff(fact_10526_ATP_Olambda__321,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_gw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_321
tff(fact_10527_ATP_Olambda__322,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_hk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_assn_assn(times_times_assn(aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_322
tff(fact_10528_ATP_Olambda__323,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_gy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_323
tff(fact_10529_ATP_Olambda__324,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun677759971260770784VEBT_o(aa_fun5749144123862529590VEBT_o(aTP_Lamm_ga(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_VEBT_VEBT_complex(Uua,Uuc)),aa_VEBT_VEBT_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_324
tff(fact_10530_ATP_Olambda__325,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_gm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_325
tff(fact_10531_ATP_Olambda__326,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: fun_Co1900716531650368264omplex,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun144181944155979022eger_o(aa_fun9138836748425859551eger_o(aTP_Lamm_gi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_Cod5621350732091595593omplex(Uua,Uuc)),aa_Cod5621350732091595593omplex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_326
tff(fact_10532_ATP_Olambda__327,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_gg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_327
tff(fact_10533_ATP_Olambda__328,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_fy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_328
tff(fact_10534_ATP_Olambda__329,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_gk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_329
tff(fact_10535_ATP_Olambda__330,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_gc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_330
tff(fact_10536_ATP_Olambda__331,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_go(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_331
tff(fact_10537_ATP_Olambda__332,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ge(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_332
tff(fact_10538_ATP_Olambda__333,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_gq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_333
tff(fact_10539_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_kk(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_334
tff(fact_10540_ATP_Olambda__335,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_dm(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_335
tff(fact_10541_ATP_Olambda__336,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_dn(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_336
tff(fact_10542_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_lx(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lw(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_337
tff(fact_10543_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_nq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Uu))
        & ! [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,numeral_numeral_nat(bit0(one))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I2)),X_13)
            <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X5: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),set_VEBT_VEBT2(Uua))
             => ~ ? [X8: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X8) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X5)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_338
tff(fact_10544_ATP_Olambda__339,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_jc(Uu),Uua) = Uu ).

% ATP.lambda_339
tff(fact_10545_ATP_Olambda__340,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_VEBT_VEBT(aTP_Lamm_jd(Uu),(Uua)) = Uu ).

% ATP.lambda_340
tff(fact_10546_ATP_Olambda__341,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_jg(Uu),Uua) = Uu ).

% ATP.lambda_341
tff(fact_10547_ATP_Olambda__342,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_jh(Uu),Uua) = Uu ).

% ATP.lambda_342
tff(fact_10548_ATP_Olambda__343,axiom,
    ! [Uu: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_jb(Uu),Uua) = Uu ).

% ATP.lambda_343
tff(fact_10549_ATP_Olambda__344,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_jf,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_344
tff(fact_10550_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_ja(Uu),Uua) = Uu ).

% ATP.lambda_345
tff(fact_10551_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_je,Uu),Uua) = Uu ).

% ATP.lambda_346
tff(fact_10552_ATP_Olambda__347,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ar,Uu) = Uu ).

% ATP.lambda_347
tff(fact_10553_ATP_Olambda__348,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_as,Uu) = Uu ).

% ATP.lambda_348
tff(fact_10554_ATP_Olambda__349,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_an,Uu) = Uu ).

% ATP.lambda_349
tff(fact_10555_ATP_Olambda__350,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ao,Uu) = Uu ).

% ATP.lambda_350
tff(fact_10556_ATP_Olambda__351,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_am,Uu) = Uu ).

% ATP.lambda_351
tff(fact_10557_ATP_Olambda__352,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_352
tff(fact_10558_ATP_Olambda__353,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_353
tff(fact_10559_ATP_Olambda__354,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ax,Uu) = zero_z3403309356797280102nteger ).

% ATP.lambda_354
tff(fact_10560_ATP_Olambda__355,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ay,Uu) = zero_zero_complex ).

% ATP.lambda_355
tff(fact_10561_ATP_Olambda__356,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_au,Uu) = zero_zero_real ).

% ATP.lambda_356
tff(fact_10562_ATP_Olambda__357,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_at,Uu) = zero_zero_rat ).

% ATP.lambda_357
tff(fact_10563_ATP_Olambda__358,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_jj,Uu) = zero_zero_complex ).

% ATP.lambda_358
tff(fact_10564_ATP_Olambda__359,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jk,Uu) = zero_zero_real ).

% ATP.lambda_359
tff(fact_10565_ATP_Olambda__360,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_av,Uu) = zero_zero_nat ).

% ATP.lambda_360
tff(fact_10566_ATP_Olambda__361,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_jl,Uu) = zero_zero_int ).

% ATP.lambda_361
tff(fact_10567_ATP_Olambda__362,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aw,Uu) = zero_zero_int ).

% ATP.lambda_362
tff(fact_10568_ATP_Olambda__363,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fo,Uu) = one_one_nat ).

% ATP.lambda_363
tff(fact_10569_ATP_Olambda__364,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_fp,Uu) = one_one_int ).

% ATP.lambda_364
tff(fact_10570_ATP_Olambda__365,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_fq,Uu) = one_one_int ).

% ATP.lambda_365
tff(fact_10571_ATP_Olambda__366,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_kv,Uu)
    <=> $false ) ).

% ATP.lambda_366
tff(fact_10572_ATP_Olambda__367,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_lj,Uu)
    <=> $false ) ).

% ATP.lambda_367
tff(fact_10573_ATP_Olambda__368,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_lk,Uu)
    <=> $true ) ).

% ATP.lambda_368
tff(fact_10574_ATP_Olambda__369,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_cj,Uu) = suc ).

% ATP.lambda_369
tff(fact_10575_ATP_Olambda__370,axiom,
    ! [Uu: $o] : aa_o_fun_nat_nat(aTP_Lamm_ck,(Uu)) = suc ).

% ATP.lambda_370
tff(fact_10576_ATP_Olambda__371,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cl,Uu) = suc ).

% ATP.lambda_371
tff(fact_10577_ATP_Olambda__372,axiom,
    ! [Uu: int] : aa_int_fun_nat_nat(aTP_Lamm_cm,Uu) = suc ).

% ATP.lambda_372

% Helper facts (1)
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X8: real] : aa_real_o(P,X8) ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(snga_assn_VEBT_VEBTi(xa,x)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),numeral_numeral_nat(bit0(one)))))),x))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,na))),numeral_numeral_nat(bit0(one)))))),xb))),pure_assn(xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,na)),xa),xb)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,na)))),xc)) ).

%------------------------------------------------------------------------------